Robust Multi Variable Control

8/14/2019 Robust Multi Variable Control

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3R o b u s t M u l t i v a r i a b l e C o n t r o l

O s c a r R. G o n z f i l e zDepa rtment o f Electrical and

Com puter Engineering,Old Dominion University,Norfolk, Virginia, USA

A t u l G . K e l k a rDepartment o f Mechanical

Engineering, Iowa StateUniversity, Am es, Iow a, USA

3 .1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .10373 .2 M o d e l i n g . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .10373 .3 P e r f o r m a n c e A n a l y s i s . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .1039

3.3.1 MIM O Frequency Response and System Gains 3.3.2 Perform ance Measures3 .4 S t a b i l i t y T h e o r e m s . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . .1042

3 . 4. 1 N y q u i s t C r i t e r i a 3 . 4. 2 S m a l l G a i n C r i t e r i a

3 .5 R o b u s t S t a b i li t y . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . .10423 .6 L i n e a r Q u a d r a t i c R e g u l a t o r a n d G a u s s i a n C o n t r o l P r o b l e m s . . .. . . .. . . .. . . .. . .. . . ..1043

3 .6 .1 L i n e a r Q u a d r a t i c R e g u l a t o r F o r m u l a t i o n 3 .6 .2 L i n e a r Q u a d r a t i c G a u s s i a n F o r m u l a t i o n

3 .7 H ~ C o n t r o l . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .10443 .8 P a s s i v i t y - B a s e d C o n t r o l . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .1045

3 . 8. 1 P a s s i v i t y o f L i n e a r S y s t e m s 3 . 8 .2 S t a t e - S p a c e C h a r a c t e r i z a t i o n o f P R S y s t e m s 3 . 8 .3 S t a b i l i t y o f P R S y s t e m s 3 . 8 .4 P a s s i f i c a t i o n M e t h o d s

3 .9 C o n c l u s i o n . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .1047R e f e r e n c e s . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .1047

3.1 Int roduct ion

A m a t h e m a t i c a l m o d e l o f t h e p h y s ic a l p r o c e s s t h a t n e ed s t o b ec o n t r o l l e d i s n e e d e d f o r c o n t r o l s y s t e m d e s i g n a n d a n a l y si s . T h i sm o d e l c o u l d b e d e r i v e d f r o m f i rs t p r i n ci p l es , o b t a i n e d v i a s y s te mi d e n t i f i c a t i o n , o r c r e a t e d b y o n e e n g i n e e r a n d t h e n g i v e n t oa n o t h e r. R e g a r d l e s s o f t h e o r i g i n o f t h e m o d e l , t h e c o n t r o le n g i n e e r n e e d s t o c o m p l e t e l y u n d e r s t a n d i t a n d i ts l i m i ta t i o n s .I n s o m e p r o c e s s e s , i t i s p o s s i b l e t o c o n s i d e r m o d e l s w i t h a s i n g l ei n p u t an d o u tp u t( S I S O ) f o r w h i c h a v a s t a m o u n t o f l i te r a tu r e i sa v a i la b l e . I n m a n y o t h e r c a s e s, i t i s n e c e s s a r y t o c o n s i d e r m o d e l sw i t h m u l t i p l e i n p u t s a n d / o r o u t p u t s ( M I M O ) . T h i s c h a p t erp r e s e n t s a s e l e c t io n o f m o d e l i n g , a n a l y si s , a n d d e s i g n t o p i c s f o r

m u l t i v a r i a b l e , f i n i t e - d i m e n s i o n a l , c a u s a l , l i n e a r , t i m e - i n v a r i a n ts y s t e m s t h a t g e n e r a l i z e t h e r e s u lt s f o r S I S O s y s t e m s . T h e r e a d e r i se n c o u r a g e d t o c o n s u l t t h e r e f e r e n c e s a t t h e e n d o f t h i s c h a p t e r f o ra d d i t i o n a l t o p i c s i n m u l t i v a r i a b l e c o n t r o l s y s t e m s a n d f o r t h ep r o o f s t o t h e l e m m a s a n d t h e o r e m s p r e s e n t e d .

3 .2 Mode l ing

T h e d e r i v a t i o n o f a m a t h e m a t i c a l m o d e l f o r a l in e ar , t i m e -i n v a r i a n t s y s t e m t y p i c a l l y s t a rt s b y w r i t i n g d i f f e r e n t ia l e q u a -

Copyright 2005 by AcademicPress.All rights of reproduction n any form reserved.

t i o n s r e l a t i n g t h e i n p u t s t o a s t a n d a r d s e t o f v a r i a b l e s , s u cl o o p c u r r e n t s a n d t h e c o n f i g u r a t i o n v a r i a b l e s ( e . g . , d i s p l am e n t s a n d v e l o c it i e s ) . I n t h e t i m e d o m a i n , t h e d i f f e r e ne q u a t i o n s c a n b e w r i t t e n a s f o l l o w s :

P ( 7 9 ) ~ ( t ) = Q ( 7 9 ) u ( t ) , ( 3 .1 )

w he re 79 a__d / d t i s a d if f e r e n t i a l o p e r a t o r ,u ( t ) E T4m i s a v e c to ro f i n p u t s , ~ ( t ) E 7 -4 i s a v e c t o r o f th e s t a n d a r d v a r i a b l e s c ath e p a r t i a l s t a t e v a r i a b l e s , a n d P ( 7 9 ) a n d Q( 7 9 ) a r e d i fe n t ia l o p e r a t o r m a t r i c e s o f c o m p a t i b l e d i m e n s i o n s . T h e v eo f o u t p u t v a r i a b l e s i s, in g e n e r a l , r e p r e s e n t e d b y a li nd i f f e r e n t ia l c o m b i n a t i o n o f t h e p a r t i a l s t a t e v a r i a b l e s a n d

i n p u t s :

y ( t ) = R ( 7 9 ) ~ (t ) + W ( D ) u ( t ) , ( 3 .2 )

w h e r e y ( t ) E R Pi s t h e v e c t o r o f o u t p u t s a n d w h e r e R ( 7 9) a nW ( D ) a r e d if f e re n t ia l o p e r a t o r m a t r i c e s o f c o m p a t i b l e d i ms i on s . T h e s y s t e m r e p r e s e n t a t i o n i n t r o d u c e d i n e q u a ti o n s a n d 3 .2 i s c o m m o n l y r e fe r re d t o a s t h e p o l y n o m i a l m a tdescription ( P M D ) a n d i s t h e m o s t n a t u r a l r e p r e s e n t a t io n fm a n y e n g i n e e r i n g p r o c e s s e s . F o r a n a ly s i s a n d d e s i g n , a s ts p a c e r e p r e s e n t a t i o n t h a t i s a n e q u i v a l e n t r e p r e s e n t a t i o n

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3 Robus t Mu l t ivar iab le Cont ro l

I d ' I do

FIGUR E 3.1 ClassicalUnity Feedback Configuration

for a l l in i t ia l condit ion s x(0) . Th e test for asy mp totic s tabil i tyis that the eigenvalues of A have negative real parts . Asym ptoticstabil i ty implies BIBO stabil i ty, which is a property of anexternal representat ion. For BIBO stabil i ty, only the poles ofa transfer func tion ma trix need to h ave negative real parts .

Consider now the classical unity feedback configuration inFigure 3.1, where r( t) E ~/ ' deno tes a vector of referenceinpu ts and wheredi( t) E Tm and do(t) E T~p denote vec torsof d is tu rbance inpu ts a t the inpu t and o u tpu t o f the p lan t,

respectively. Assume that the state-space representations(transfer function m atrices) of the plant an d controller aregiven by (A;, B; , Cp, Dp)(Gp(s))a n d ( A o Bo Co D c ) ( G c ( s ) ) ,respectively. The re pres enta tion o f the closed system willbe w el l -fo rmed i f the d imens ions( @ ( s ) E ~ ; ( s ) ; man dGc (s) ~ 7Ep (s) rep)are compat ib le and i ff l + DpD~[ ~ O.T hestate representat ion of the closed-loop system is of the form :

2d (t) = Aclxd(t) + Bd w(t) , y( t) = Cclxd(t) + D dw (t) ,(3.9)

wherexd( t ) = [xp( t ), &( t ) ] Tan d w (t) = [ r( t ) , do (t) , di(t )] T.The proper t ies o f the dosed - loop sys tem are de te rminedby analyzing (Acl( t) , Bcl( t ) , Cd(t) , Dd(t)) .For example, theclosed-loop system is asym ptotical ly s table if the eigenvalueso f A d ( t)have negative real parts . The d ose d-loo p system is thensaid to be inte rna lly stable.

The analysis an d design of comp lex feedback configura tionsis s implif ied by using the general dos ed-lo op block diagram inFigure 3.2, wherew ( t ) is the vector of a l l exogenous inputs ,y~c(t)is the vector of controller outputs ,z( t ) is the vector ofperfo rma nce variables, a nd u/(( t) is the vector of inputs to thecontroller. The top block is cal led the two-input and two-ou tp u t p lan t , P, and the bo t tom one cor responds to the matr ix

of controllers denoted by K that is formed after a l l the con-trollers have been pu lled ou t o f the closed-loop system. As a

W

YKP(s)

K(s) 1

b Z

- - U K

FIGURE 3.2 Ge neralClosed-Loop Block Diagram

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trivial example, the classical unity feedback is represented ithe general block diagram with the following transfer functiomatrices:

uK(s) J

where:

[ I - I - @ ( s ) : - @ ( s )

P ( s ) = = / 0 . . . . 0 . . . . . . 0 . . . . i . . . :. . . .

Le2 ' ( s ) i P22(s) ] L I - I -G p(s ) : -G p(s )

In this example,K(s) = Gc(s ) ,the exogenous inpu ts vec to ris w(t) = [r( t) , do(t) , di( t )]r, the vec to r o f per fo rman ce var i -ables has been taken to bez ( t ) = J r ( t ) - y ( t ) , y1((t)] T,a n dthe vector of controller 's inputs and outp uts is s imply given buK(t) = uc(t)a n d yK(t ) = yc( t ) ,respectively. The first per-

. , Aformance variable xs the tracking error,e ( s ) = r ( s ) - y ( s ) .A s ta te -space represen ta tion o f the two- inpu t and two- ou tpuplant is the following:

ke ( t ) = A pxv( t ) +[0 0 Bp:B P LyK(t) j (3.10)

L u : ( t ) Jz ( ' t ) l = . . . . . 0. . . . . .Ly /d t ) ]

L - C p j L 1 0 - D p ! - D ~ J(3.11)

The c losed- loop t ransfe r func t ion matr ix f rom the exogenous inputs to the performance variables is as follows:

z(s) = Tzw(S)W(S).

The closed-loop transfer function matrix is in fact a lowelinear fractional trans form ation (LFT) ofP(s) with respect toK ( s ) as given by:

Tzw(S)= S rg ( P ( s ) , K ( s ) ) = P l l ( S )

+ P 1 2 ( s ) K ( s ) ( I - P 2 2 ( s ) K ( s ) ) - 1 p 2 1 ( s ) .(3.12)

Based on this LF T, the general c losed-loop system is welform ed if ]I - P22(oc)K(oc)] ~ O. In the tr ivial examp le of c lassical unity feedback system, this reduces to [I- P22(0cK ( ~ ) I = ] i +D~DcL ~ o.

3 . 3 P e r f o r m a n c e A n a l y s i s

The purp ose of the co ntroller is to m ake the closed-loop systeme et the desired specif icat ions regardless of the unc ertain tpresent . Examples of specif icat ions are that the closed-loosystem shou ld be asym ptotical ly s table , the s teady-state erro


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fo r each o u tp u t channe l shou ld be sma ll , and these s tab i l ity andpe r fo rm ance spec if i cat ions shou ld be m a in ta ined n o t on ly fo rthe no mina l p lan t m ode l bu t a l so fo r a ll mode ls in a spec if i edunc er ta in ty set . In th is case, the c losed - loop system wil l be sa id tohave the p rope r t i e s o f robus t s t ab i l ity and robus t pe r fo rmance .

Con sider the c lass ica l uni t y feedback configu ra t ion in Figure3 .1 wi th an add i t iona l senso r no i se vec to rr l ( s ) so tha t Uc(S) =r ( s ) - ( r l (S ) + y ( s ) ) .The p e r fo rm ance analys is i s depen den t onthe fo l lowing th ree types o f t r ans fe r func t ion ma t r ice s :

Retu rn rat io : L o ( s ) = G p ( s ) G c ( s ) . Sensi t iv i ty : S o ( s ) = ( I + L o ( s ) ) - 1 . Complementa ry sens i t iv i ty : To ( s ) = L o ( s ) ( I + L o ( s ) ) - 1 .

These t rans fe r func t ion m a t r ice s are de f ined when the loopis b roken a t the ou tp u t to the p lan t . S imi lar de f in i t ions fo l lowwhen the loop i s b roke n a t the p lan t 's inpu t . The inp u t /o u tp u tand inpu t /e r ro r r e la t ions can be wr i t t en a s shown nex t :

y(s ) = So (s )do (s ) + So (s )Gp (s )d i (s ) + To (s ) r ( s ) - To (s )r l (S ) .(3.13)

e(s ) = -So (s )d o (s ) - So (s )G p(s )d i (s ) + ( I - To (s ) ) r ( s ) + To(s )~ l (S) .

(3.14)

No t ice tha t the m app in g f rom the re ference inpu ts , r ( s ) , tot h e e r r o r s c o n t r ib u t e d b y t h e m ,e r ( S ) , is g iven byI - To ( s ) .Inth is case, i f the c lose d- loop system is in ternal ly s tab le , thes teady -s ta te tr ack ing e r ro r co n t r ibu ted by a vec to r o f s tep re fe r-ence inpu ts r ( s ) = Ro ~ , Ro E 7 -4is foun d u s ing Laplace 's f ina lv a l u e t h e o r e m t o b ee r( cx D ) = ( 1 - To ( O ) ) R o .Thus , the e r ro rcon t r ibu t ion s to each o u tp u t channe l wi ll be ze ro i f in add i t ionto in ternal s tab i l i ty,To ( O ) = I .Because the sensi t iv i ty and c om -plem enta ry sensi t iv i ty t ransfer fu nct io n ma tr ices sa tisfy :

S o ( s ) + To ( s ) = I , (3.15)

t h e n S o ( O ) = O p presults in e~(oo) = 0. A sufficient co nd itio nfo r the dc ga in o f the sens i t iv i ty ma t r ix to van ish i s tha t eve rye n t r y o fL o ( s ) must have a t leas t one pole a t the or ig in . This isthe gene ra l iza tion o f sy s tem type to M IMO sys tems . Fu r the r-more , i f S o ( O ) = 0 p x p , the s teady -s ta te e r ro r con t r ibu ted by avec to r o f s tep func t ions a t the ou t pu t d i s tu rbances ,d o ( s ) , will

a lso be zero . An addi t io nal co nd i t ion is needed fo r zero s teady-s ta te e r ro r con t r ibu ted by a vec to r o f s tep func t ions a t theinpu t d i s tu rbances ,d i ( s ) . A suff ic ien t condi t ion is tha t theplant has no entr ies with poles a t the or ig in .

Mak ing T (0 )= I r e su l ts in the des i red ze ro s teady -s ta tee r ro r s to vec to r s o f step func t ions a t the re fe rence inpu ts anda t the inpu t and ou tpu t d i s tu rbances . Th is cho ice , howeve r,has the un des i rab le e ffec t o f mak ing the e r ro r con t r ibu ted bythe dc com po nen t o f the senso r no i se no t to be a t t enua ted .Th is is a co m m on t rade -o ff in con t ro l sy s tems , wh ich does no ta ffec t the des i red pe r fo rmance a s long a s the s igna l - to -no isera t io fo r low f requenc ies i s ma de su ff ic ien tly h igh .

Ze ro s teady -s ta te e r ro r is poss ib le fo r s tep inpu ts by ap p rop r ia te ly inc lud ing an exog enous mo de l o f the exosys tem in thfeedback loop (Gon zalez and Antsakl is , 1991) . I f zero s teads ta te e r ro r i s no t needed , then i t w i l l be necessa ry to makthe mapp ings f rom the fou r exogenous inpu ts to the t r ack ine r ro r in equa t io n 3 .14s m a l l . Three ways to de te rm ine the s izeo f a t r ans fe r func t ion ma t r ix a re p re sen ted in the fo l lowinsubsect ion .

To s imp l i fy the p re sen ta t ion , a s sume f rom n ow on tha t thplant is square with p = m. In addi t io n , s ince the physica l uniu sed fo r inpu t and ou tpu t s ignal s m ay lead to e r ro r s o f d i ffe reorders o f mag nitud e , i t is usefu l to norm alize or scale thmagn i tudes o f the p lan t 's inpu ts and ou tpu ts . P rocedu res tpe r fo rm sca l ing o f MIM O sys tems a re p re sen ted in , fo r examplSkoges tad and Pos t lethwa i te (1996) . One app roach i s to noma l ize the p lan t' s inpu ts an d ou tpu ts so tha t the magn i tud e each e r ro r is le ss than one . An a l t e rna t ive and co m m on cho ii s to inc lude the no rma l iza t ion in the f requency -depende

we igh ts to be in t rodu ced fo r con t ro l sy s tem des ign .

3 .3 .1 M I M O F r e q u e n c y R e s p o n s ea n d S y s t em G a i n s

To de te rmine thef r e q u e n c y r e s p o n s eof a BIBO s tab le t rans fe rfunc t ion ma t r ix ,G ( s ) , l e t it s inpu t be the vec to r o f complexexponen t ia l s u ( t ) = u _ e J ~ t , u _ E C P ;then , the s teady-s ta tere sponse i s a complex exponen t ia l vec to r o f the same f rquen cy wi th am p l i tudes and phases changed byG ( S ) l s ~ j ~ o .Let the s teady-s ta te response be g iven byy s / t ) : y ' e J ~ t ,

y E CP, then the c om plex vectors _u and y are re la ted by:

y = G ( j t o ) u . (3 .16)

T h e c o m p l e x m a t r i xG ( j t o ) E C p pis ca l led the f requencyresponse ma t r ix , and i t c an be u sed to de te rmine the s ize oG ( s ) at a par t icu lar f requ ency to .

In genera l , the s ize ofG ( s ) is def ined as the ga in f rom aninpu t to i ts co r re spond ing ou tp u t . I f the inpu t i s u ( t )= _ueTM,the ga in o fG ( s ) at to can be def ined a t s teady-s ta te to be theratio of the Euclidean vector norms [[y[[/][_u[[.T h i s c o n c e p t o f

gain is bounded as fo l lows:

i f ( t o )= r a i n I IG ( j t o ) u l l ~ I IG ( jt o ) _ u l l ~ma x I I G ( j t o ) _ u l l _ ~ ( tJr_~ll~0 I1_~11 t l~ , I tII_~ll~0 I 1 ~ 11

(3.17)

wh ere 6-( to) and ( to) a re the la rges t and smalles t s ingulvalues of G ( j t o ) . These bound s a re u sed to de f ine the s ize oa TFM as follows:

Large G ( s ) is said to be l a r g e at co i fff (G (j to )) is large . Small G ( s ) is said to bes m a l l at to if 6"(G(jto)) is small.


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typ ica lly added on the exogenous inpu t channe ls and theper formance ou tpu t channe ls . The fo rmer weigh ts se rve toinc lude in the model spec t ra l in format ion about the inpu ts .The la t ter weights are important in design to emphasize thef requency bands where the per fo rman ce ou tpu ts need to beminim ized. A possible design problem that results is to f ind ap ro p e r c o mp e n s a to rGc(s)so tha t II Tz,o[l~ < 1. This a nd oth ercontrol p roblem s will be discussed in the following sections.

3 . 4 S t a b i li ty T h e o r e m s

In this section, a brief introduction wil l be given to variousstabil ity cr i teria that can be used to determ ine the s tabil i ty of aclosed-loop control system.

3 . 4 .1 N y q u i s t C r i t e r i a

Con sider a negative feedback interconn ection of a plant G p(s)and controller G o(s) in F igure 3.1. LetnLo+d e n o te th e n u m b e rof uns tab le po les o f the re tu rn ra t ioLo(s) .Then the Nyquis ts tabil i ty cr i ter ia is given by the following theore m:

Stability Theorem :The closed-loop system consist ing of thenega t ive feedback in te rconnec t ion o fGt,(s) a n d G,-(s)is inter-nally stable if and only ifnLo+ = ncp+ + nG c+and the Nyquis tplot of 1I + Lo(s) Iencircles the originnLo+t imes in the anti-clockwise direction and does not pass through the origin.

The f irs t condit ion guarantees that the closed-loop systemhas no uns tab le h idden modes . T he second con di t ion g ives aMIM O genera liza tion o f the Nyquis t p lo t in te rms o f thed e t e rmin a n t o f I + Lo(s) .Because of the de term inant, theNyq uist plot of a scalar t ime sI + Lo( s )is not simply a scaledversion of the N yquist plot of [I +Lo(s)[ .

3 . 4 . 2 S m a l l G a i n C r i t e r i a

An othe r cr i ter ia that is often used to d eterm ine internal s tabil-i ty o f the feedback in te rconnec t ion i s the sm al lgain theorem,which is based o n l im it ing the loop gain of the system. Thistheor em is central to th e analysis of robust s tabil ity. The smallga in theorem s ta tes tha t i f the feedback in te rconnec t ion o f

two proper and stable systems has a loop-gain product lessthan unity, then the closed-loop system is internally s table .There exist several versions of this theorem. One version isgiven next .

Theorem:Con sider the feedback system in Figure 3.1, wheresystems Gp and Gc are proper and stable . Then, the feedbacksystem is interna lly stable if:

IIG/l llGcll < 1

The stabil i ty theorem s already given can be used in theanalysis and synthesis of control systems. These theorems arealso useful to de term ine co ndit ion s for robust s tabil ity and

O s c a r R . G o n z f il e z a n d A t u l G . K e l

perform ance o f the closed-loop system with the real plant adiscussed in the following section.

3 .5 R o b u s t S t ab i l i ty

Controller design uses a nominal plant model. The erro

between the nominal model and the real plant ar ises primarilf ro m tw o so urc es : u n m o d e le d d y n a m ic s a n d p a ra me t r i c u ncertainties . If the co ntroller design does no t take these errorin to account , i t cannot guaran tee the p er fo rmance o f thclosed-loop system with the real plant n or g uarantee thathe closed-loop system will be s table. Therefore, i t is im po rtanto design controllers that w ill main tain closed-loop stabil i ty isp ite o f e r roneous des ign models and uncer ta in ties in parameter values. Controllers so designed are said to im part s tabil itrobustness to the closed-loop system.

To analyze stabil ity robustness, con sider the un ity feedbacsystem in Figure 3.1. A basic plant uncertainty representat iois G;( s ) = Gpo(S) +A~(s), where G;o(S) i s the nomina l p lan tmodel and where Aa(s) is the addit ive uncertainty. Otheuncer ta in ty represen tat ions inc lude the ou tp u t mul t ip lica tivuncertainty, Gp(s) = ( I + Ao( s) )Gpo(S) ,and the inpu t mul t i -plicative one,Gp(s) = Gpo(S)( I + Ai( s) ).For design purposes,i t helps to normalize the un certa inty representat ions. Foexample, consider that the real plant is represented with aou tpu t mu lt ipl icat ive uncertainty, and le t Ao(s) =Wo ( s ) A o ( s ) ,where A o ( s ) a n d Wo ( s ) are proper and stable with[I/~o(s)[[~ < 1 and whereWo ( s ) i s a f requency-dependen tscaling matrix . In this case, the u nity feedb ack system in Figur

3.1 will be ro bus tly stable if an d o nly ifI IToWo[I~


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3 Rob ust M ult ivariable Control

X(s) I~

W

FIGURE 3.4Pulled O ut

[ I M(s) b Z

Simplified Closed-Loop S ystem with Uncertainties

usefu l for rob ustne ss analysis. In F igure 3.3, /~(s) is the b lockd iagona l mat r ix o f a l l the normal ized uncer ta in ty b locks([IA(s)]]oo _< 1), andPo(s) i s the augmented nomina l p lan t ,inc lud ing the uncer ta in ty f requency weigh ts .Po(s) is assumedto be s tabilized byK (s ) . Us ing a lower LFT, the bo t tom loop inFigure 3.3 can be closed result ing in Figure 3.4, whereM ( s ) isproper and stable . IfM ( s ) i s par t i t ioned cor responding tothe two ve ctor inputs a nd outp uts and if ][MH ]Ix < 1, thenthe closed-loop system is robustly s table . These results thatmake use o f the uns t ruc tu red uncer ta in t ies lead to very con-servative results. One way to reduce the conservativeness is totake advan tage o f the s t ruc tu re in the b lock d iagona l m at r ix/~(s) as don e w ith the s tructure d singular values.

3 .6 L i n e a r Q u a d r a t i c R e g u l a t o r a n dG a u s s i a n C o n t r o l P r o b l e m s

3.6.1 Linear Quadratic Regulator Form ulation

The l inear quadratic regulator (LQR) is a c lassical optimalcon t ro l p rob lem used by many con t ro l eng ineers . So lu t ionsto LQ R are easy to com pute and can typ ica lly be used tocom pute a base line des ign usefu l fo r compar ison . A fo rmula-t ion o f the LQ R prob lem cons iders the s tate equa t ion o f theplant:

2 ( t ) = A x ( t ) + B u ( t ) . (3.22)

The f orm ulatio n also considers the following quad ratic costfun ction of the s ta tes and control inpu t:

ts

] = x T ( t f ) S x ( t f) + ~ x r ( t ) Q x ( t ) + u T ( t ) R u ( t ) d t ,(3.23)

0

where S = S r _> 0, Q = Q r > 0 and R= R T > O.To min imize the cos t func t ion , cons ider the Hamil ton ian

system with s tate and costate(p( t ) ) dynam ics g iven by:

[~(t)]p ( t )

with ini t ia l condit ions:

1043

- Q - Lp(t)]x(O) = Xo; p(tf) = Sx( tf) .(3.24)

The optim al controller uses full-sta te feedback and is giveby :

u (t) = - R 1 B T p ( t ) x ( t ) = - K ( t ) x ( t ) ,

where P(t) is the s olution of the m atrix Riccati equation:

P ( t ) = - P ( t ) A - AT P ( t) + P ( t ) B R - I B T p ( t ) - Q ,

which is solved backward in t ime start ing atP( t f ) = S .T heopt imal feedback ga in matr ixK( t ) is given by:

K = R - 1 B r P ( t ) ,

and the optim al cost is as fol lows:

J o p t ~ - xT(o)P(O)x(O) .

3 .6 .2 L i n e ar Q u a d r a t ic G a u s s i a n F o r m u l a t i o n

The l inear quadratic Gaussian (LQG) control problem is aop t imal con t ro l p rob lem where a quadra t ic cos t func t ion imin imized when the p lan t has random in i t ia l cond i t ionswhi te no ise d is tu rbance inpu t , and whi te mea surem ent no isThe typ ica l implem enta t ion o f the LQR so lu t ion requ ires thathe plant s ta tes be est imated, which can be posed as an LQGproblem . The plant is described by the fo llowing state anoutp u t equa t ions:

2 ( t ) = A x ( t ) + B . u ( t ) + B w w ( t) .

ym(t) = Crux(t) + v(t)(me a s u re me n t o u tp u t ).

yp( t ) = Cpx( t )(per fo rmance ou tpu t ) .

(3.25)

T he v( t ) a n d w ( t ) a re uncorre la ted ze ro-mean Gauss iannoise processes; th at is,w ( t ) a n d v ( t ) are wh ite noise processesw ith c ovariance s satisfying:

[0v ( t ) ] [w r ( t +"r),The quadra t ic cos t func t ion tha t i s to be min im ized i s g ive

by:

i ]= E x T ( t f ) S x ( tf ) + ~ x T ( t ) Q x ( t ) + u r ( t ) R u ( t ) d t ,0

(3.26)


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1044 Oscar R. Gonzdlez and Atul G. Kel

where S= S T ~0 , Q =Q r _> 0 an d R= R T > 0 . The op t im alcontroller is a ful l-sta te feedback controller an d is given by:

u ( t ) = - K ( t ) 2 ( t ) ,

where 2 ( 0 is the K alm an state est imate. The closed-loop stateequa t ion can then be g iven by:

Ix( t ) ]= A d ( t ) [ e ( t ) J+Bc ' ( t ) [wl : l ] 'where:

A c, B u ,t ,] 0 ]0 A - G ( t ) C m ' B d = Bw - G ( t ) "

The closed-loop state covariance ma trix is as fol lows:

~ - ~ ( t ) = A c l (t ) Z ~ ( t ) + Z ~ ( t )A ~ ( t ) + B c l ( t) [ WO ]B T ( t ) ,

where ~ ( t ) = [x ( t ) ,e(t)] r, and:

[ x ( O ) x r ( o ) x ( O ) e r (o ) 1y ] ~ ( O ) = E L e ( O ) x r ( O ) e ( O ) e T ( O ) j .

The per formanc e ou tpu t covar iance is g iven by :

and the inp ut co variance is given by:

, , [ - - K T ( t ) ]Z u ( t ) = [ - K ( t)K ( t ) ] Z ~ ( U [ K T ( t ) J .

Th e co st is as follows:

, = ~ T r [ ~ 0 0 ]K - "' t '+ f[Q+KT(t)RK(t) --KT(t)RK(t)IZ-~ tf ' J[ KT"(t)RK(') KT (t)RK (,)Z s , ( ' ) d t

RemarksFor conven ience o f im pleme nta t ion , the s teady-s ta te so lu t ionobtaine d in the l im it as tf---* oo is used. The LQ G co ntroldesign is optimized to reject white noise disturbances; how-ever, i t can be modified to handle constant dis turbances viafeed-forw ard and integral control . The selection of gains inthe feed -forwa rd case can be d one in a s imilar way as in thecase of tracking system design. Prior to im plem entatio n, therobustness of LQG designs needs to be evaluated since there isno guarantee that any useful robustness wil l be obtained. Forconstant dis turbance rejection via integral control , one needsto use integral Kalman f i l ter and also integral s ta te feedback.

The integral LQG can be used both for dis turbance rejectioand tracking. Similar modificat ions are possible to handltracking of t ime-v arying reference inputs .

3 .7 H ~ C o n t r o l

The LQR and L QG can a lso be posed as two -norm op t imization prob lems (referred to as H2 control problems). If thop t imiza t ion p rob lem is posed us ing the Hoo norm as thcost function, the H~ formulation results .

T h e H ~ control problem can be def ined in te rms of thegeneral c losed-loop block diagram in Figure 3.2 where thexogenous signals are include d in the vector co(s) and aapprop riate ch oice of perform ance variables is given by thvec tor z(s) . The TFM P(s) inc ludes f requency-dependen tweights and appropriate normalization as described previously. C onsider the following realizat ion o f P(s):

2 p ( t ) = A x p ( t ) + [ B l ! B 2 ] [.w . t.).] (3.27 )I r K ( t ) ]

.Z.( . t ) . ] {.C12.]Xp(t)_}_ 0 i a ,2 1U K ( t ) j= ( /: 5 2 ; i . 6 . . j [ ~ K i : ~ ] , (

where:

D21BT= 0. (3.29)

D21DT = I . (3.30)

D Tc 1 = 0 . ( 3 . 3 1 )T TD 1 2 D 1 2 = I . (3.32)

The control objective is to design a feedback controller thainternally s tabil izes the closed-loop system such that thoc-n orm of the ma pping Tz~ i s bounde d:

( IT z w i l~ = s u p I ) z ( t) l l~ < Y. (I]w(t)[12~0II ( t ) 11 2

A subopt imal con t ro lle r sa t is fy ing the above me nt ioneobjective exists if positive sem idefinite solutio ns to th efollow ing two Riccati e qua tions a re possible:

P( t ) = P ( t )A + AT p( t ) - - P( t ) (B2B T - y -2B1SW )p( t ) + C

(3.34)

0( t ) = AQ(t ) +Q (t )A T - Q( t ) ( cT c2 - , I -2CT C1)Q( t) + BT.

(3.35)

M oreover, the so lutions P(t) and Q(t) sat isfy the following

p ( P ( t ) Q ( t ) )< y2. (3.36)


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3 R o b u s t M u l t i v a r i a b l e C o n t r o l 1045

The Hami l ton ian sys tems co r re spond ing to the two Ricca t iequa t ions can be ob ta ined s imi la r ly to how th ey were ob ta inedfor the LQG case . The Hoo contro l ler, which sa t is f ies them e n t i o n e d b o u n d , i s g i v en b y:

k c = A c ( t ) x c ( t ) + B c ( t ) u K ( t ) .

yK( t ) = C c ( t ) & ( t ) .(3.37)

The ma t r ice sA t ( t ) , B c ( t ) ,and C~(t) are given as follows:

A c ( t ) = A + ~ / - 2 B I B T p ( t ) - B 2 B Y 2P ( t)

- [ I - y - 2 Q ( t ) P ( t ) ] ' Q ( t ) c Tc 2 .

S c (t ) = [ I - y - 2 Q ( t ) n ( t ) ] - l Q ( t ) C T.

C o ( t ) = - - B T p ( t ) .

(3.38)

In p rac t ice , the s teady -s ta te so lu t ion o f the Hoo con t ro lp rob lem i s o f ten des i red . The s teady -s ta te so lu t ion no t on ly

s imp l if i es the con t ro l le r imp lem en ta t ion bu t a lso rende rs ac lo sed - loop sys tem t im e- inva r ian t , s imp l i fy ing the robus tnessand performance analys is . For the s teady-s ta te Ho~ contro lp rob lem, a subop t ima l so lu t ion ex i s t si f a n d o n l y i f th efo l lowing condit ions are sa t is f ied . The f i rs t condi t ion is tha tthe a lgebra ic Riccat i equat ion:

0 = PA + AT p - - P (B 2B T2 - - "y -2B 1B ~ )P +c Tc 1 ,

has a posi t ive semidef in i te so lu t ion P such tha t [A- (B2B T -y - i B ~ B T ) p ] is s tab le . The second condit ion is tha t the a lge-bra ic Riccat i equat ion:

0 = A Q + Q A r - Q(c2Tc2 - "y -2cTC1)Q+ B1BT,

has a posi t ive semidef in i te so lu t ion Q such tha t [A - Q(CeTC2- y - 2 c T c 1 ) ] is s ta b le . T h e t h ir d c o n d i t i o n is p ( P Q ) < ,/ 2.

3 .8 P a s s i v i t y - B a s e d C o n t r o l

Pass iv i ty i s an i mp or tan t p rope r ty o f dynam ic sys tems . A largeclass of physica l sys tems, such as f lex ib le space s t ruc tures w ithco l loca ted and comp a t ib le ac tua to r s and senso rs , c an be c la s-s i f ied as be ing natura l ly pass ive . A pass ive sys tem can berobust ly s tab i l ized by any s t r ic t ly pass ive contro l ler despi teunmode led dynamics and pa rame t r ic unce r ta in t i e s . Th is im-po r tan t s tab il i ty cha rac te ri s t ic has a t tr ac ted mu ch a t t en t ion o fresearchers in the co ntro l of pass ive sys tems. This sec t ionpresents se lec ted def in i t ions and s tab i l i ty theorems for pass ivel inear sys tems.

3 . 8 .1 Pa s s i v i t y o f L i n e a r Sy s t e m s

For f in i te -d imensional l inear, t ime-invar ian t (LTI) sys tems,

pass iv i ty is equivalent toposit ive realnessof the t r ans fe r

funct ion (Newcomb, 1966; Desoer and Vidyasagar, 1975The concept of s t r ic t posi t ive rea lness has a lso been def inein th e l i te ra tu re an d is c lose ly re la ted to s t r ic t pass iv i ty.

L et G(s ) deno te a p x p ma t r ix whose e lemen ts a re p ropera t iona l func t ions o f the co mplex va r iab le s . TheG(s ) is saidto be s tab le i f a ll i ts e lements are analy t ic inRe (s) >_ O.L etthe con juga te - t r anspose o f a com plex ma t r ix H be deno te

b y H H.

Definition 1A p p ra t iona l ma t r ixG(s ) is said to be positive real (PR) if ."

All e lements ofG(s ) are analy t ic inR e(s ) > 0 G(s ) + GH (s ) > 0in R e ( s ) > 0 , or equivalent ly :

Po les on the imag ina ry axi s are s imp le and have nonnegat ive-def in i te res idues

G( j t o ) + GH ( j t o ) > 0for to E ( -- oo, c~)

Var ious de f in i t ions o fstrictly positiverea l (SPR) sys tems

are found in the l i te ra ture (Kelkar and Joshi , 1996) . Givebelow is the def in i t ion of a c lass of SPR systems: marginal ls t r ic t ly, and posi t ive-rea l (MSPR) systems.

Definition 2A p p ra t iona l ma t r ixG(s ) is said to bemar ginally strictlypositive real(MSPR) i f i t is posi t ive rea l and the fo l low ing it rue :

G(j t o ) + GH ( j t o ) > 0 fo r t o E( - o c , o c ) .

De fin i t ion 2 (Joshi an d G upta , 1996) g ives the least res tr icive class of SPR systems. IfG(s ) is MSPR, i t can be expressedas G ( s ) = G l ( s )+ G2(s), where G2(s) is weak SPR (Kelkarand Joshi , 1996) an d wh ere a l l the p oles o f G1 (s) a re p ureimag ina ry ( Josh i and Gup ta , 1996 ).

3 . 8 .2 S t a t e -Sp a c e C h a r a c t e r i z a t i o n o f PR Sy s t e m s

For LTI sys tems, the s ta te-space charac ter iza t ion of posi t ivreal (PR) cond i t ions re su l ts in the Ka lman-Y akubov ich -Pop(KYP) lemma. In Lozan o-Leal and Joshi (1990) , the KYlemma was ex tended to WSPR sys tems , in Jo sh i and Gup t

(1996) , it was ex tende d to MS PR systems. These ex tensions ag iven next .

Le t (A , B, C , D) d eno te an n th -o rd e r m in ima l r ea l iza t ion the p p t r ans fe r func t ion m a t r ixG(s ) . T h e f o ll o w i ng l e m m athen g ives the s ta te - space ch a rac te r iza tion o f WSPR sys tem.

T h e L o z a n o - L e a land Joshi (1990) Lemma:T h e G(s ) isWS PR i f and on ly i f the re ex i s t r eal ma t r ice s : P = p r > 0P E R '~n, L ~ R px'~,an d W E R pi ' ,such tha t :

AT p + PA = - L T L .

C = B T p + W T L . (3.39)

w Tw = D + D T.


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1046 Osca r R . Gonzd lez and Atu l G . Ke lk

In these equa t ions ,(A, B, L, W)is contro l lab le and observableo r min ima l , and F ( s ) = W + L ( s I - A ) - I B i s m i n i m u mphase . I f G(s) is MSPR, i t can be expressed asG(s ) = Gl ( s ) +G2(s) , where Gi(s) is WSPR and a l l the poleso f G l ( s ) a re pu re ly imag ina ry ( Josh and Gup ta , 1996 ) . Le t(A2, B2, C> D) den ote an n2th-order min ima l rea l iza t ion o fG2(s), the stable part ofG(s) . The fo l lowing lemma i s anex tens ion o f the KYP lemm a to the M SPR case .

T h e J o s hand Gupta (1996) Lemma:I f G(s) is MSPR, theree x is t r ea l m a t ri c es : P = P r > 0 , P E R~ n , g C R p< an dW E R pp,such tha t e qua t io n 3 .39 ho lds wi th :

L = [ 0 p n l , " ~ p x n 2] , ( 3 . 4 0 )

wh ere (A2 ,B2, 9 , W ) is min ima l and F(s ) = W +L ( s I - A ) - I B = W + 58 (s i- A2)-lB2 is m i n i m u m p h a se .

3 . 8 .3 S t a b i l it y o f P R S y s t e m sThe s tab i l i ty theo rem fo r a f eedback in te rconnec t ion o f a PRand a MSPR sys tem i s g iven nex t.

LM I f o r m o fPR Lemma:A n a l te r n a te f o r m o f K Y P L e m m acan be g iven in t e rm s o f the fo l lowing Linea r Ma t r ix Inequa l i ty(LM I). A system (A, B, C, D) is said to b e PR i f it satisfies:

A r p + P A P B ] + C U < 0B r P [0 7 ] r [ W rW ]( 0 C 7 ] (3 .4 1 )

P = P r > 0, (3.42)

where U = 0, V = 0, and W = - I . Th is LMI cond i t ion iconven ien t to u se in the ca se o f check ing PR-ness o f MIMOsystems. This LM I is a specia l case of d iss ipa t iv i ty LMI (Kelkand Joshi, 1996).

Stability Th eoremThe c lo sed - loop sys tem cons i s t ing o f nega tive feedback in tec o n n e c t i o n o fGp(s) an d Go(s) (Figure 3 .1) is g lobal ly asym p-totically stable if Gp(s) is PR,Gc(s) i s MSPR, and non e o f thepu re ly imag ina ry po le s o fGc(s) is a t ransmiss ion zero of Gp(s)(Joshi and Gupta , 1996) .

No te tha t in the theo rem sys temsGp(s) an d Gc(s) can bein te rchanged . Some non l inea r ex tens ions o f these re su l t s aa lso obta ined in Is idor iet al. (1999) . Pass iv i ty-based contro l-l e rs based o n these fundam en ta l s t ab i li ty re su l ts have p rove n be h ighly effec t ive in robust ly contro l l ing inherent ly pass i

l inea r and non l inea r sy s tems .Most physica l sys tems, however, a re not inherent ly pass iv

and pass iv ity -based con t ro l me th ods cann o t ex tend d i rec t ly such systems. For example , unstable sys tems and acoustsys tems a re no t passive . One poss ib le me tho d o f mak inthese nonpass ive sys tems amenab le to pass iv i ty -based con t ris to pass ifythem us ing su itab le compensa t ion . I f the com pensated sys tem is ensured to berobust ly pass ivedesp i te p lan tunce r ta in t i e s , i t c an be robus t ly s tab il i zed by any MSPR cotro l le r. In Kelkar and Joshi (1997) , var ious pass if ica t ion tec

G ( s ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G ( s ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

U

G ,(s ) l JG p ( S ) Y m = Y

(A )Series

G ( S )

Gfb(S)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a

(B ) Feed back

~ - y m = y

G ( s ), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G ( s ), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

G A s )

Gn(s) ~ Ym

(C) Feed-Forward

+ f _ ~ G p (S ) ~ y

iGf b(S ) G s (S )

1

, ~ Y m

(D) H ybridFIGUR E 3.5 Methods of Passification


11/11

3 R obu s t M u l t i va r i ab l e C on t ro l 1047

n iques a re p re sen ted , and some num er ica l examples a re g iven ,dem ons t ra t ing the u se o f such techn iques . A b r ie f r ev iew o fthese me th ods i s g iven nex t .

3 .8 .4 Pa ss i f ic a t i o n M e th o d s

The fou r pass i fi ca tion me th ods in F igu re 3 .5ser ies , feedback,

feed-forward , and hybr id pa ss i f ica t ionare g iven in Kelkarand Josh i (1997) fo r fn i t e -d imens iona l l inea r t ime- inva r ian tnonpass ive sys tems as shown. Once pass if ied , the sys tem canbe con t ro l led by any MSPR o r weak ly SPR (WSPR) Con t rol ler( Is idor i et al., 1999) . In Figure 3 .5, the sys tem with in putu ( t )a n d o u t p u t y m ( t )( G ( s ) ) represents the pass if ied sys tem. Thetype o f passi f ica t ion to be u sed depends on the dynam iccharac ter is t ics of the un pass if ied p lant . Fo r example , thesystem having unstable poles wil l require feedback pass if ica-t i o n , w h e r e a s t h e s y s t e m h a v i n g n o n m i n i m u m p h a s e z e r o s( i .e . , having unstable zero dynamics) wil l require feed-forwardpass i fi ca tion . Some sys tems may requ i re a com bina t ion o f thebasic pass if ica t ion methods . For SISO systems, the pass if ica-t ion p rocess i s ea s ie r than fo r MIMO sys tems . The rea son i stha t fo r SISO sys tems , on ly the phase p lo t needs to be checkedto de te rm ine pass iv i ty, whe reas in the ca se o f MIM O sys tems ,the KYP lemm a cond i t ions have to be checked . One num er ica ltechn ique tha t can be u sed to check the KYP lemma i s l inea rma t r ix inequa l i ty (LMI) -based PR cond i t ions . The so lu t ion o fthe LMI can be done u s ing the LMI too l box in MATLAB o rano the r semide f in i te p rog ramming package .

One imp or tan t th ing to be no ted he re i s tha t , in the case o finhe ren t ly pass ive sys tems , the u se o f an MSPR con t ro l le r

gua ran tees s tab i l i ty robus tness to unmode led dynamics andpa rame t r ic unce r ta in t i e s ; howeve r, in the ca se o f nonpass ivesys tems tha t a re r ende red pass ive u s ing pass i fy ing compen-sa t ion , s t ab i li ty robus tness dep ends on the robus tness o f thepass i fi ca tion . Tha t i s , the p ro b lem o f robu s t s t ab i l i ty i s t rans -f o r m e d i n t o t h e p r o b l e m o frobus t pass i f i ca t ion .In Kelkarand Josh i (1998) a num ber o f su ffic ien t cond i t ions a re de r ivedto check the robus tness o f the pass i f ica t ion .

3 .9 C o n c l u s i o n

This chap te r has p re sen ted some o f the fundam en ta l too l s inthe ana ly s is and des ign o f linea r, t ime- inva r ian t , co n t inuo us -

t ime , robus t , mu l t iva r iab le con t ro l sy s tems . The chap te r s ta rwi th an in t roduc t ion to mode l ing . The ana ly s i s too l s inc lubas ic m easu res o f pe r fo rmance , f r equency response , and s tb i l i ty theo rems . L inea r quad ra t ic , H~ and pass iv i ty -basecon t ro l syn thes is t echn iques w ere a l so in t rod uced .

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