n91-, 1206 - nasa · means is needed to preveul the experimem from driftin,,_ inlo ils own...

N91-, 1206

MI(3{OGRAVITY VIBRATION ISOI,ATION: An Ot'TIMAI, (',ONTI_.OI, LAW

h_r Like ON I',-D1M F,NSIONAI, ('.ASE

R.ichard I). Ilampton

University of Virginia

Thornton Ilall, Mc('ormick Road

Charlotl,esville

VA 22901

Carlos M. (; rodsinsky

NASA Lewis R,csearch Center

Mail Stop 23-3

CI(,vcland

OII 44135

Paul E. Allaire, David W. Lewis, Carl R. Knospc

University of Virginia

l'hornton llall, Mc(',ormick Road

( ',harlol, l,csville

VA 229111

413

https://ntrs.nasa.gov/search.jsp?R=19910011893 2020-05-09T11:19:04+00:00Z

:\ B S'T 1,_A( _"1"

('ert,aiu exp,,rinlout, s contemplated for space platfort_s tt;_t_t I., _ ]>_,;,I,t_,,t l'tL.n,

tile accelerations t,f the platform, in this paper au optimaJ active (:out, rol i

deveh)pcd for microgravity vibration isolation, using constant state feedback gainl_

(i(tentical l,o those ()l)l,_dx_(,d from the l,inear Quadrati(. Regulator [LQR] ai)l)V( ach,

along with constant fee(lforward (preview) gains.

The (tuadrati( cost function ['()r this ('ontrol algorith_ (,lfectiv(qv _.i..,]kI.,.

oxl,erm_l a('('eleralitms of the platform _list, url)anwes I)y a fact,or l)toporliu_at t,J

(l/w) 1 I.t)w rre(luOttcy a('(:el(;ratAonts (h,ss t,hau 50 Hz tu'e .tlt, miate(t bv _,t, _+t+,t

titan t.wo ordors of ma_t_itu(le. The ('ox)tro[ relies on the at)solut(, l)o._it,i()tL a_+i

veh)citv [o(,dl)a(.k o[" th(, ex,,rim(,1_t and the absolute po,,qtion and v.l(Jcilv

fcvd-l()rv,,,azd ()f _1_(' 14.tfc)run. atL(l gonerally (leriv(,_ the _t_abilitv rot)u>:tl_,_:.

chaxa(t(,rist i('s ,_l,n:_nt('o<t I)v th(' I,QI{ ,l>l)loa.(h t()Ol)t_i]ua]it.,,;.

+lh,, [t_t+tho, i _s ,h,tivo_l is extotl(lal)l(+, to the ca,_(' it_ whi('h t_u_lv tho r_4ati',(,

l>osiliOtlS all(l \ol{)<'il it's attd lho al)soltlt_' a('coh'rations of th(, ('x[)('[illi_'lll all(i s[);l(,'

l>lat, lt)rnl are a.vailal)lo.

414

1. INTI_ODUCTION

A space pla.Lform experiences local, low frequency acceleratiot_s (0.01-:It) l tz)

(lue to equipment motions and vibrations, and to crew activity it]. Certain

experiments, such as the growth of isotropic crystals, require an (mvironment in

which tile accelerations amount to only a few micm-g's [2]. Such an environment is

not presently available on manned space platforms.

Since the experiment and space I)latform centers of _favit y do not coincide, a

means is needed to preveul the experimem from driftin,,_ inlo ils own ¢>rbital molion

and into the q>ace l)lal ['Of'Ill wall. Additionally, sonm <',:perimenl,s require umlglical_

to f_rovide l>ower, experiment c_mtrol, coolant flow, communications linl,:z_o, of

otl, r services. I_nforl_umtely. such t_easures also mean that unwanted l>l_torl_

accelerations will I,e trallsmitted to the experiments, l'his necessi_ales exp<imen[

iscflation. I'assiw' is_lators, however, cannot coml)ensale for umbilical sl.iffness, nov

can they achieve low _,_ouch corner Ir<,quencies even if umbilicals ar_' absenl.

Active isolation is therefore essenlial.

The pmhlen_, lhel|, is tO design m, aclive iso[alioll svste[ll tO mini_,,dze these

imdesired accetera, ion t ransmissi_ms, while aclth,vin_ a<leqxlale slabilily Imtr_ins and

system rot)ustness. Spatial and <o_,l rol energy linfitalions n_ust also t,,

a ('('()in ii1o(1_tt e(-[.

415

2. MATIIEMATI('.AL MODEL

The ._eneralI)roblemhas thrc_,translational and three rotatiCm_tldegr<,esof

fre('(lon_. For simplicity, however, this analysis will (onsi(l('r oI_l_ 1.1_(,

one_tinwnsioual probh'm. The general problem could be treated il_ an analogous

maturer, l,et the experiment be modeled as a mass m, with position ×(t).._\,_sume

that the space station has position d(t), and that umbilicals with stiffness k and

d_ttnl)ing c conm,ct the experiment and space station. Suppose further that a

iI_a_neli(" actuator appli('s a. ('Olltrol force proportional lo the appli(,(t (:utrecht i()).

wilh l)rOl)Ort.ionldil v (:onsl_lJtl <_<.Such a K)_odel is shown ill l:igure I.

The syste))) ('(lu_)ti())_ ()f)))()tion is

lltX ÷ C(.X--<'f)+ k(x--(1) 4 _)i = 0 ,!

l)ivisiotl I)v I11 at|(I t_'at'ral_g('lll('llt yi('l(ls

._ = k (x-(t)- (' (._-<'1)- ++i t"-l)-S i_ ))-i

[I_ state.' space _oI _,tt ion thi_ becon_es

W h(q(?.

: .\ x +)2u + f (3

x'tf't f 't

A ,: I< (I 0

, 12= c_-- [_

,116

u=i ,f=_ {°tkd+C_

The objective is to minimize the acceleration x(t).

417

:_. OPTIMAL ('()NTI{()I_ PROBLEM

'File optimal control problem is that of determinin_ lho

)l(t) --- i whi('h minimizes a. :uitable performance index

'OI/l IO[ ('11 ['I'OI] l

.I = J(x, u, 1,) (,1)

l'oz the sysWz)) des(zib('(l by Eq)). (3) subject to the state variable c(>z)di)ioa,_

--()

[ i Ill X(t ) _-<(_) (51j)

:\n()ther rcas(>nal)h, _ssutlq)t.ion is tl,at t_(t) is Ixml_(l(,(I. a,_(l it will b(, fi./l_,i

tuath(,mali('ally a(Ivai)la._(,()its (ai_(l ()ulv)tfiuiulall.v i(,._i rictiv(,) ).() _LS:lLIllCt,hal l_! I i-

I i I_l 1-(1)= L) • i( i

:\ (tm_(lrati_ l)('rfor_mmcc indt:x

.I=.1/ ix I'WIx+ w:lu- ](ll,-0

:118

has I)('(,)[ chosen, _ls oue that, lends i(,._(,If well to the radar iom(I ai)pr()ach t.o Ol)i il_)_

lh( ,,i)i)(,[' lizlfi( ,.)f (,he (lefi][il(,('(mtrol:. since an amdyl.ical s()luIio_ is desired. '" '

iutegral has been :(,h,('ted so a._ t t) yiel(I a time-invariau( c()ntroller, ller(, W ! i: ,_

square 2x2 (onsta)_t w('i_htin_ matrix wlfile w:l is tt w('i!_,hl.iug coll,'..;131tl.

Although, \V1 couhl bea full 2x2 matrix, for this probleln a ,liagonal l'ort_t

has been employed for the sake of simplicity.

[w 0 ]WI = la

0 Wl b

r]'ll{' perfornliuR:e index cottseqllently reduces to

I[ ''x" 2 . 2 ".I = q ) [WlaX [ 4 WlbX 2 + w:lll-]dl

-. {)

so lhal _'a.('h ::tare is weie, hu,d imlepend_,lll.ly.

If sinusoidal motion o1" lhe experit_ont is considered, so lhal.

x(I,) --: I_ si. _¢

,)

alld i.i(l) = _.'-x(t i, lhe cost ['utwtion can I)e exlm,sse_l iJi 1_,:]3_,,,cd ltle ;l.cc_,]_,vat}_ii

;|lid ('01111_] /-I,'q

I {>':; wla -+- w lbl 1'_2'_:+ + ,.,,':lu23 ,_l_.I--:3 ( I --qi, ¢9,

It is apl>ar<,nt thal ll+is l.,_,rlorljlalw_, il++]ex co+_,,:,_,l+J_,lll],+,.,,+,i_}lt:-;;t<_._,]+,r;tti,.+t+ ,_1 ,,.,

fre,+t_w'twi+"sI_ul<'h mot<, 1ha, at, higher freqttett,'+h,s.

ORIGINAL PAGE IS

OF POOR QL_ALITY

_19

.1. SOI,('TI()N

Fin(liilg the optimal control to minimize Eqn. (4) is a variatioli_tl l)rol)iem of

l,agrange, for whi('h the iuitial steps of the solution art, well-kll()wli i('._,., l']lbert

[4]). The va.riatiomd aDl)roach is out, liiled below, fi)llowillg which t lw ,()l_l)li(_,I lolls

a(hh'd I)y the nonhom()g('neous t.erm [(t) will I)e addressed. (',lrre,_) Ol)i.i))_al

controls texts either assume that _f(t) - [) (e.g., [4], t). 262) or ro<tuire that it l_av(_ _

restri(:te(! range space (e.g., [6], p. 238). The solutiou that follows l)rovi(h:,s aH

amtlytieal ()l)lh_al with¢)ul imposing such restrictions.

The a.i'_lliil(,iit ()t" the (:osl tum:l.ion .l t'r()ul !((11_. (4) is alt_lll,'l_r_'(I 1)v ll_,,

l.a_ralt_¢' mllltil)li('r A tilu(,:_ the svsteili e(liJation ()f motiou E(I u. (;11 wt_('r¢,

'\ 1A= A.,I(J

lhe result .J cart I)(, ,,xt)r('ss['_l a._

= li _[II)

wl_er_' lhe llamiltol_ia, u il is

I!

= _ "' , ATtt _ (x r \v 1 × + w:l_') _- (_- ..\x- L,,_-{)

It is (l('sire(l i.o ot)t_dtl an ol)limal s()luti()u u = u

Th,' Iirsl. variali()ti (,f J(x, li. _) is

which lliiililliiz('s .J.

1'2

t:J = ['_' foil _)It 011J

420

which is set -qual _o zero to minimize, J. l]owever, imoarating by parts,

j'0( J_ ",.

ooi) l l _) dt = _ ,__1_. dtilx 0

so that, the al) _ve expression for b.l be(x)mes

/ij }.c,x,[ 0II= ,laH _ "_'r)_._+ _,}dt- 0• 0

(13)

l'h)l h bx _md bu a.r(' a.r/)itrary ,,,'a,ria, ti(ms, s() LJ = 0 only if

81! "Ti l4a)

1 tl)_

lhe conditions ,zi,.en I)y It(In. (5) Sl ill apply.

SOlving '[.tins. (l'ta) and (I,11)) yields

_=wt x-A__ (15a!

* 1 TIt ...... I) ..\

v,.3 - _(1,51)

oy Poor (_u/_rw42I

Temporarily elimin;Lting u t)roduces the result

._ x f(1(;)

W hOl'("

A

-A T

If I"qu. {1(;) is uo_ s,)lv_'_! I,_r ,__in l,,rl_l._ ,fix _n_l 1:. I&llL. ( 151)ii will II_,l, tll_i.-.l_ ;L_

exl)re,_sion for the oI)lilLlal ('onl,r()l u .

..\s n()l,e(I I_el'orp. Ol)limal (()lltrol I.OXIS _ouerally tl('a.[ the homoa_,neous

pml)lem (where f(l) _ ()), but lhry do uot provide _tll _tualylical s_lution _o l tl_.

nonhomogt!neous ,_vstelll _lesclilwd by (5) and (16). ,S_dukvadze has lreat,,d t)l_.,

nonht)mog.n_'OlLS l_I'ol_h,_ [.l,5j, t)ul. his _lifl'i('ult ll'(';ll, lIl('tll ,_'¢'IIIS l_vgeI5* _,, h_w

remaiut'd either uu_co_l)rel=ende_l or _nutl_'r-al)l)reciat_'tl. This lne ho¢l is _'_:!_,._i,_i!'.

well-suiuxl Io low-fre_lUe]_c.ydist.url)_mct,rejection,and h_,_b_'_'uapl>li_'_ll_t.h_w,_

t,l_(? I)['(,,,,;CI_Ii)rol,l('n_.

The homo_,or_eottssolutionto Eq.. (15),where f = LI,is

I ,l ,l (17_

422

ORIGINAL PAGE IS

OF POOR QUALITY

The four eigenvalues of A may be found to be, in ascending order of real parts.

-/71 + (/_l 2- 4_2 )1/2(18a)

,)-;71 - (;fl" - 432 )1/2

'2

ilL'

(tsh)

IL3 = - iL1(iSc)

#1 = -H2(IS(l)

where 171 and ;}'2 are defined as follows:

and

2k31 = __m

(' "2 a w 1 b

in'2 m w 3

,)

39 = ;Jl'-,1 ,) + --_" in-w 3 lii"

19a)

l!)b)

The eigenvect, ors of A corresponding to the respective eigenvalues itk n]ay be chosen

to be

l

Irk

") 2 ._l(.,t2 + it.k )7.1 'l + z _Pk = __ + ---Irk _31tk '?3

"t1 + ('72 + Irk)It k

(20a)

423

where 71 , 7.2, 73, and 74 are defined below:

(20b)

(2oc)

,}

7,3 =: 9-Ill W3

(20d)

'I Ia(20e)

l:sing Eqns. (18) through (20) with (17) the solution to the homogeneous sv,,,,,,iK: is

i_1w. llgt -It 1t -ltgt ]

cle El + c'2e_ P.> + c3e-lt, l t -i_.2 t

h { "2 "'2 P:'2 +cle PZl2]

(21)

t Ekt I ' k = 1..... .1 alltl where c I ..... c I at,' /trt)it:arv c_)ustauts.with 12k = L Pk., J

.,\ppiica, lioll ill' the vitriltl.iOll ill' t);t.I'a, IIII'I_TS IlJellu)d wi:l_ Iel'lliilLa] _rJ¢lit]ol:>

(Eqns. 5b.c) leads to the gOllOl'it.i solution of t,he iio11- tllllOg('llO011S S}'SI,{'_I/I, wit h tx_o

constants of integration yet undetermined.

If the two constants of integral ion are eliminated by solving for _ in l erms :>f

x and [, the general solutions for k I and '\2 become:

424

-1_ 1t --ttgtAl (lXi + (,>x,> + (3 e + "4 (22a)

-ll. 1t -1,.2 t"\2 = _sXl + (6x2 + s_7e + so8e (22b)

ORIGINAL PAGE ISOF POOR QUALITY

in which tile _i's are functions of the eigenvalues and eigenve(:tors ()f :\, and of tile

disturbance .f(t).

The Solution Form

Using the fact that

* ]. '_

u (t)= _.3 AIb__ [cf. Equ. (151))] (_:3)

the optimal control is found to be

. q:}o-/.q tat t -l_2t f elt2tu (t) = qlXl + ,l.2x.2 + , 'Itl f.,(I)dt + q,1e f.)(t)dt

where

(2-1a)

--m kql = --7 (m - ltllt2) (24b)

--[11 C

02 =--_(_+1_ I +It. 2) (21()

m t 2 (: k (21(1),/:)= 'a (_)(/'t + _/q + _)

I 9 k_ rn,_ )(.,.). (,q4 a _ _ +_lL',+_) (_,h,)

(It should be. notod that the [eedba('k _a.ins ql and tl.2 are lhose wili('it w<_uid

result from applying standard LQi{ theory to the hoJnogene()us >3s_ez_ _'(tua,'ion

= Ax + bu). In gqns. (24) ltl, it,2 are the eigenvalues of :\ with ne.g_ttive real

parts, [see Eqns. (18a,b)] and

0_" POOR" Q_UALWY

425

f,)(t) = k ,1 + c' (_Ill Ill

(24f)

By tel)eared application of tile method of integration I)y parts, tile control may be

re--expressed in terms of an infinite sum:

1,,r+l + q4 r_-O /;.)r$i ]LL(t) = qlXl + q2x2 + q3 "r=O

(25)

I{(,writiu_ f.) in terms (ff d iul(I (1. the (:ontr()l function I)e('omes

4¢

+ E (-t) i-I (' ' " )i k ,___qq3 ;/1 ,1 (t(i)(i=1 _( +- )+(-1 _ __7_T+It I It.2 1t ] _.) [-_r{)_

+ [(--l)n-I (: , q:_ q,l )l (1((---_ + n "' n)(t) + hi_h,'r ol<h'r _'rm,_

ltl #2(_,(i)

11

This lila.y I)e wril, len in a tllor(' al)l)('alill_ I'orill as

u (t)=c x(t)+("p v .{:(l,) + ('dO d(t) _- (:(11 (t(t) + Ifigh(,r orcl_,r t,erm:

(27

in whi(:h the constant coelTiciem, s ('1)' ('v' ('(lO' and C(l[ Jn_v I)e del'ine(l t'ro_L t:_(ln._

(24) and (26). harly, if the int'init(, stuns converge rapidly ,,hough, The Ol)tilnal

('ontrol can be apl)roximaJ(,d hy

426

.,g

u (t) = Cp x(t) + cv _:(t) + cdO d(t) + ('dl c'l(t ) (28)

For v(_ry low frequency disturbances tile higher or(l('r t(,rms itl 1";(t11. (25) at-(,

negligibly small, and the control (Eqn. (2¢)) closely approximates the opt, imal. If.

in fat:t, the second- and higher-order derivatives of d{t) are i(lenti(',lly zero, th('

apl)roximation is exact. It can t)_ shown that for the critit:ally damp(,(l closed loop

s3sl._m the eigeuvalues are real alld equal, and that th(, ('owvet',4(,t_c( _ _s _xt()t(, _al)id

than for the overtlamp(,d system. Further, as the clos(,(t--lt_op :vst_,m t,i,_oHv_.tlu(,_

t)4'c(_l_e mor_' uegativ(: l,h(, C()IlVCr_OI|('e S])(_()(l ff,()CS lip _l.,q w(,l[.

427

5. CONTROL EVAIAJATION

Physical Realiz.a.bility of the Control

The control, Eqn. (25), is physically realizable, if the states and sufficien,

derivatives of d(t) are accessible (or estimable by an observer), and if the higheL

order terms are negligible. It is not necessary that the eigenvalues be real. alt hough

the proof of this re(iuires a more general linear-algebra or sta c-lra.,tsition-t,_atrix

approach.

If values a,_, a:_signed to the systent parameters, associated controller _ains

can I)e evaluated. Suppose t,hatm = 100 Ibm, k = 0.3 lbf/ft, c = 0 Ibf-sec/fl. at_ct

+_= t0 lift/Amp. With w 3 arbitrarily set at 1 and Wlb vari 'd. associated z te,-,._,

vahtes of Wla can be found below which the eigenvalues it I ,,nd 1_.2 will +ilwavs b_:

real. Such values atre tabulat,ed in 'l'al)lo 1. Stated otherwise., tho lal,tlal¢,¢l ',alt_¢,s

of the weights Wla and Wlb are those integer values (for the _ak,, of sittq_licity) lot

which t,he closed loop system is closest, to I)eing critically {l.tltq}e<t ,,it hour bein,g

underdamped. Corresl)onding controller feedback and t'eed-(orward _uins (l¢)r t+h_,

first five derivatives) are also included.

The states x(l,) and x(l,) and tit,, _h,rivat.ives ¢ltq_)(l). ,l'l)ll/ ,tll_[ +[(21(t) t_,

clearly availal)h, for a)_ ('ill'lh---])i|se<[ s_.sl(qll. [l(jw(W(,l. ill .-,l,_l(('. lilt, ()llIv /t[,5()l[lVl,

measurements which can be dir('ctly a.va.ilal)h+ are, x(1.) and d(t), itot_t wl,icI_ xiw).

('l(t) and x(t), d(t) are obtainal)le only by successive integration(s). |{earran_,cm,,tlt

of (28) into

or

u (l) = (cp + cdo)x(t) + (c v

u (t) = (Cp + Cd0)d(t) + (c v

+ c.ij )x(t) - c +c>{..<(t, -,I(_ ]-,.,:++,r_cti - d(t)]

+ ,. ilj,'t(t,) + cp[x(t)-dC, )] + Cv[i(t)- a(,)]

-.-, ,

30 )428

obviates the need for one accelerometer, but one accelerotucter plus two iutegtations

remain necessary for either the platform or the experiment. Since [x(t)--(t(t )] (or

one of its integrals) h_ not been weighte(l in the performa/_ce index J, expt,tmmnt

drift will be a problem that must be corrected either I)y another control loop or by a

change of system states. The latter could be accomplished by incoq)orating an

accelerometer attached to the experiment into the. state e(tuatiot_. .\ltetnatively.

oue could append a u integrator to the plant, include the ('urr(,l_¢ i(t) +ts a t l,it<l statue.

and optinfize the contr()l di/dt. But fur the sake o1 simplicit,v (i.+... f(,w,,r .,lm(,s) 1t_(_

f(>tZll_'r )la.s })e(_ll i-ts,'.;IJlil('<l {wil, holt| (levehqml(,/_t) in 1his paper.

The higher order terms of the control [E(lns. (25) and 2_;)] can b(' m,%i('('t(.,+i,

for h)w fre(luencies, if the eigenvalues it l and It2 at(' of sufficient _nod_+l_:. lhr,s,,

eigenvalues, in turn, are under the (:ontro[ of the ([e: i_ner, d(_t(utltim,(l I)v t_i: , tlui((,

of woigllts Wlit, Wll), and w3. It is al)l)ar(,llt frolll ,'.ql]. (25) 111;1t tl Ii) (,_.,(,!_ti;tllv

le(hl+(_s to two alteltm, titLg ])()wer :eries.l,'or a sintlst)idal <lisl_trl);_ti((' (ff It('(lli(_:v _

the :eri(,s form of the t'<mtrol ('(mver ges for +#//_+i+ < I (i = 1.2). It (au be silowti

,X. )r ;+', 2 r ,,that each alternating power series converges like *.L' (-1 ,-) . With +'lowr=f) 'It

fte(llletlcy distt+rban(:es (i.e., small re.lative t() systen+ clos(,d loop ,,]a+,nv:_l:+_,s) +t

(t)l+Irol ['orttl('([ I)v series t, rtltt('a.ti(m vmv ('[os(,lv al)l)roxitttalos t t_(, ()l)t i_.al.

lPor examph,, sut)l)OS[ + t]la! I]1(! _(>rt_mlized ft'('qu_,_(i('s 1.+'//_il i_)r ;t ._it _l_-(+i(I,tl

disturl)att(:e are less than 1/5, and that only the feedfot'ward (t)tlttol t(,tttt._ { !0(ll:

and (dl(](t) are inclu(h,d with th(, fe(,(ll)at'k l.erlllS. |_Vell SO. t+h(' f_+,'(tl'()t_at(t +, rli(,t:

of the trtmcated cot_trol, at any tim(, t. will I)e a ('urrent that is still witlti_t .l , [i.<.

(1/5) 2 of the feedforward portion (>t' the actual ot)titltal. It' Ih(, ttortttalizt,(I

t're(luett(:ies are below 1/10, this <.I.l)[)roxilll+tl,}(Jql (.Tror xvi]l bo l('ss th_-tlt 1'/,,. "labl_+ 1

shows that the gains c(l i t)f higher order derivativ(,,., (lIi)(t) [see E(tn. (2(;) fur

algO)raic representations] are, in fa(-l, quite, small.

+_t,+, .:.+,,+g D_,2- ,,,+-,

429

Ill some eircumsl, ances there may be design ccmstrainls which prevent the

designer from selecting weights that will lead to sufficiently rapid convergence.

ttowever, since convergence occurs rapidly even for eigenvalues of relatively small

modulus (I,.,/Iti] < I/3), in a great many cases the (lesi_oner will have much

latitude in his choice of weights. For "low" frequency disturbances, in these cases, t_

control which includes only one or two f_dfi)rward u_rms will be "close" _o the

optimal. I he:e frequencies will be well-attenuated.

tligher frequency disturbances will also be well-attenuated, provided the

input-to--output transfer function(s) are at least strictly proper iu the l,_tplaco

Transform variaJ)h, s. This will not__._I)e the case for the, l)r_'sent t)rol)h,l_ it _ltor_' th_,r_

c >j are included in II,_'¢'¢_lllml t'lactic,_il_, ,lli._three feedforward _a.ins (cd0, Cdl, el2

means that, only i_rot)ortional and first--derivative feedforward [Eqn. (251 wilh r =

0,1 or Eqn. (26) with n = 2] should be added to the feedback control terms..,ks wilt

be seen shortly, however, adding even the proportiom_l foe_llorward term(s) can

dramatically iml)rove the (tist,ul't)al|Ce re.i('(:tiol| over l,haJ aI'[or_i_'d I)v LQI_ fe_,dback

alone.

Transfer Function and Block Diagram

Neglecting the higher order terms.

otn put accelerations or displacements is

the transfer funct[oll bet, woen [llI)ll_ amt

,_ ('

s-X(s) _ _ (77 - _'_11 ss21)'s)t = (77ts-il7.-2 --+ (7_c

+ (k_ _.¢10

+ c )s +- (17 +

(31)

and a block diagram of the controlled system can be drawn as in Figure '2.

•13o

Control Stability, Stability Robustness, and General Robustness

Since the control feedback gains are the same as those obtained by solution of

the standard Linear Quadratic Regulator (LQR) problem, the closed loop system is

stable and enjoys the stability robustness characteristics guaranteed by (,he (LQR)

approach to optimality, viz., a minimum of 60 ° phase margin, infinite positive gain

margin, and 6 dB negative gain margin [6]. Additionally, nurnericaI checks indicate

that it enjoys substantial insensitivity, or general robustness to _|ncertainties in k, ('.

and m, as indicated by Table 2 and Figures 3 through 10. By comparing the Bode

plots of Figures 3, 5, 7, and 9 (correspon(ling to controls using both LQR F/B and

proportional F/F) with those of Figures 4, 6, 8, and 10, respectively ((:orrespondil_,z

to controls using LQI{ F/I'I only), one ca. see that adding fee(l-f()rwar, l

substantially improves (listm'l)ance reje(:tion at low frequencies, t"or ('xa_@e .

('Oral)arisen of Figures :l with Figuret indicates thal th(, optimal ('onlrol iil(_thod

described above ca_ lead to acceleration reductions of ,grca,ter tha, l_ four orders of

magnitude for all frequencies. This reduction is more tha, n two orders of magnitude

below that afforded by LQI{ feedback al(me at the lower frequencies, i.e., those most

heavily weighted in the l)orfornmuce i.dex.

The order (_t' I,he r(,dut:t.ion is ('V('lltlla.llv Iimit(M [_v c()lllro[ (<)_t. ()t COIlrSO.

I)robably in terms either of a,t',tuator-lelated limitations (such as heat-rcllit)va[ or

force-generation reqtlirenlents) or of power limitations (especially in a space-statioll

environment). The control also leads to displacement reductions of lhe same

magnitude, limited in this case by actuator-stroke or spatial limitations. Providina

a unit transtnissibility for very low frequencies and weighting (xml)and/or f(x-d)

in the performance index ,I wou[d [)e steps toward ad(hessin_ these, latter

limitations.

431

Computational Aspects

A significant amount of algebra was required to solve the two-state problem

of this paper, and the labor involved increases dramatically with each additional

state. Itowever, such symbolic manipulators as MACSYMA may be used to ease

the workload if a symbolic solution is desired. Further, well-known numerical

methods exist (i.e., Potter's method [7] or Laub's lnethod [8]) for solving the

solution to the homogeneous system. These can readily provide the feedback gains

in numerical form, even for problems with many states. It _rlight be anticipated.

then, that a nunmrical m(,thod also exists for finding the desir( 1 feed-forward gains.

Such is tile case, as will Iw shown in a later paper.

432

6. CONCLUS[ONS

This paper has applied an existing method for obtaining an optimal control

to the microgravity platform isolation problem, for which the disturbances to be

rejected are low-frequency accelerations. The system was assumed Io be

representable in the form _ = A x + _b u + f, with quadratic cost function

l, oo Tj = 1.z (x WlX + w3u2)dt and diagonal weighting matrix W 1. The cosultant,0

control law was found to be simple, stable, robust., and 1)hysically realizable.

Further it was shown to have excellent acceleration- and (lisplacemenl.-altet_uation

(ha.racteristics, and to be frequency-weighted toward the low en(l of the a(celeratio_,

S[)('C(i'Hln.

The method is extendable to the case for which only relaJive positions and

velocities, aml al)so]ute accelerations, are available; and can be al)pli('(l so ,s to

weight relative (lispla(:(,n_,nts in the i)erforrnan('e index.

The approa(:h as presented is al_,ebraically

manipulators can 1)(: used to ease the algebraic ,abors.

produces feedback gains identica,! to lhose obtained

intensive, but symbolic

Further. since the method

by the LQR al)proach t()

()ptimality, numerical computation of those gains is easily ac('omplishod, even for

largo s,:s_ems. Th(, f('('(l-forward gain:_ can I)e found numerically with COml)arabl,'

e0.5c.

7. ACKNOWLEDGEMENTS

The authors wouht like to recognize NASA for partial fimdin_ of thi,_ _ork,

Eric Maslen at the University of Virginia for his many helpful control-r(_ ated

suggestions at problem points an(l regarding flmlre work, and Dr. Gerald Brown of

NASA Lewis for his insights into the _nicrogravity isolation problem in an ort)ital

environment.

433

8. REFERENCES

l. llamacher, tl., Jilg, R., and Merbold, U., "Analysis of Microgravity

Measurements Performed During DI," 6th European Symposhun ol]

Materials SciencesUnder Microgravity Conditions, Bordeaux, December

2-5, 1986.

2. Alexander, J. Irwan D., "Experiment Sensitivity: Determination of

Requirements for Vibration Isolation," Vibration Isolation Technology

(,enter, September 28-29, 19,_.Workshop, NASA Lewis Research ' ,"

3. Elbert, Theodore F. Estimation and Control of Systems. New York: Vau

Nostrand lh,inhohl Coral)any luc., 198.1. Chapter 6: "Optimal ('¢)lktml of

Dynanfic Systt,lns."

Salukvadze, M. E., "Analytic Design of Regulators (Constant.

Disturbances)," translated in Automation and Remote Control. Vol. 22, No.

10, pp. 1147-1155, October 1!)61. Originally published in AvtomaTika i

Teh,nmkhanika, Vol. 22, No. 10, 1)1). 1279-1287. February 1961.

Salukvadze, M. E., "The Analytical i)esign of an Optimal Control in the

Case of Constantly Acting Distm'l)ances," translated in ._\ummati()n and

Romote Control, Vol. 23, No. 6, Pl). 657-4;67, .June 1%2. (')ri_ir_;_llv

published in Avtomatika i Teh'makhanika, Vol. 23, No. 6, Pl). 721-7:{1. July

1!)62.

Anderson, B. I). O. and .Moore, .I.B. l,inear Optima! ('ontrol. Englewood

Cliffs, New Jersey: Prentice t|;LII, 1971. pp. 70-71.

Potter, J. E., "Matrix Quadratic Solutions," SIAM .lourtlal of Al)t>liod

Mathematics, Vol. 14, No. 3, May 1966, pp. 496-501.

Laub, A. J., "A Schur method for solving algebraic Ri(:cati ,_quations," IEEE

Transaction_ on Automatic Control, Vol. AC-24, pp. !)13-92t.

°

.

.

.

,

434

Plate

Shaker

k

C

---gsD---Actuator

////////////111/////11//I

xtt)

Figure 1. System Model

435

Tabll, 1. Optimal F/F and F/B Gains for Sele('tod StateVariable and Control Weightings.

Syst em Parameters:m = 100 Ibm k = 0.3 lbf/ftc = 0.000622 lbf-soc/ft ({ = 0.17,)(_ : 10 Ibf/aznp

t W, iKhl s ___

Wla ] wIh l 13,;:1;,,23 ;I

I41 Ill

li.1 5 I

92 fi _ I

126 7

1t15

211!1 '1

5,_t 15

IIl.l.I 2t1

11il7 2"_

2:12!1 lit I

3171 .15 I

11 13 Iq_ I

_i:I_5 i111

1ti5Xl KII

25!11 I 11111

I"/11Gauls

CP

I. 11845

3, 1:1'24

4. 7659

6. :17:12

7. !17111

'1 51117

11 PI'II)

12 _11:1

I 1 121i!t

Ili 11:121

21 u71o

Q. IJ_'l

111 I_1!1

I'_ 12!17

"_h 2_11i

t_ I I:llil

¢v

I. 116117

2.1-113

2._210

:1. 1514

I. 1552

3.7:1711

:1 !19.1!1

I. 2;I;,11

I. Ili7,1

3. 1729

li. ,120!I

[ I11i_11

7 TI.II

_, tblll

I 'lh ]:lhll III. q'_)(i

I ] 12_. 7:172 12,111Z5

I ! .....__ _ llil).!l:lX!t I | I 1417

CdO _ ] I"d 1

O. 02941 -oI

0.0297 I -41.00q11

0.0"2!1_ I 41.1x)q)(1

t). 0'2!1!1 I I1 11114)11

11.112!1!t 11.O111H1

0.112!1!1 I i1.11111/t)

11.112!.19 t1.tl41011

t1.tl2!1!1 ] I1.11111111

11.112!19 It.1tllt111

I1.11'2!1_1 11.I141tttt

tl I1:11111 11 1)11111

n

tt.1t:11111 I I1.11011 [

o .11.1191) il I10(11

l) 11.1111) II I1111J [

11 11;111tl 111111111

1111:11141 t1 tllltl I

11 I111t11 l) 1111111

Ii I):IIXI tl .1111tll

II I1 Itlt1 II 1t0411

l"/t +' li,,,L _ tl _1

1',12 (:d:)

--0.04)70 -41 04_7

_1 ttt1,111 -.11.11t.119

41. o11211 o 1)o10

-.11 Ill) 1:'i _1,111_7

_) I11112 _9 ()005

-0.t101t) 41tltt04

-41 111111X q) .11110:|

_) t1411t7 _1. I)002

-o. 0006 _). Ol)1)2

-41 0o06 -0. uO02

-o. III111,1 -4) 111)(II

-41, I|(111,I+ -1) It11111

t). 1111112 tl. 11111111

-_1 t It 111'_ -41 Ill)Ori

-II i)l)|l_ 41 II(tt)ll

4i itllll I 41 il 1t II )

1t Ill)lll 11 1191t11

111111111 _ll !1141| I

41 1111111 _t clllltll

Cd.l (',15

-0. UO,19 -0 1111:1:2

4) 0009 _1. 0110.1

q), Ol X/'| 41. l)(J() [

_1 04x12 _) 1to111

_1).11ollt 41.t)l)llq

-,11,1)11111 _I iilia)q)

_11, t)IJl) [ -4t t111ttll

-Q.tlttIII -lJ Ill tlt_ )

_J I)1t1111 _! 11111!11

-Q. ti111!1t _1 Illitll)

ql I/I11!11 41 Ill)l!ll

_11 ()()lltl it _luil9

4) lll)ltl) 41 I)lll!lt

_11 11111111 -It lillillt

_j 11111111 41 LIIitlll

41 11111111 !) I)tltllt

41 t1111111 _t ()114t11

411111t111 41 11111111

--II, Iit1111l -1) Illllllt

436C?.:7_:r,_'+,_ PAGE IS

s21)(s)

X(s) ,[_/_/ _ [

Plant

C,)ntrol l_,r

Figure 2. Blot'k Diagram

437

Table 2. Closed loop transfer functions for system with designparameter values of k = 0.3, c = 0.000622, and m = 100; butwith actual parameter values as shown. 61, G3, (;5. _Lnd (;7include both LIIR F/B and proportional F/F; G2, I;4, (;6. and G,_include LQR F/B alone. Weigtlting parameters used were

Wla = 258, Wtb = 10, w3 = 1 (see Table t).

System Parameters

_lbf_ Ib_tseck _tiT-_ c ( )

I). 3

m (ibm)

o.ooo622 ioo(<=().12)

O. 3 O. 00o622 100

O. 45 O. 000622 I00

O..15 O. 000622 100

0. 00622 tO0

O. 3 0.00622 100

O. t5 O. 00622 91)

(). 45 0.00622 90

Closed Loop l"ransfer function

s'-'D(s)

al(s) - 0.0000622s + ().0001.)

O. 31056s"-4. 4675s _ 16. Ot)2 t

G2(s) =

63(s) =

0.0000622s + 0.(_3()0

0.31056s2+4. 4675._+ lb. 0 ?2

0.0000622s + 0.0151

0.31056s2+4.4675s+lG.OTT l

0.1)t)00622s + ().()450(;4 ( s ) - .)

(}.31(}56s-+4.4675_+lb.(}TT t

t;5(s) -0.000622s + 0.00()!

.)0.31056s "+4. 4680s*16. 062 t

G6(s) -0.000622s + 0. 0300

0.31056s2+4.4680s+16.o62 t

(;7 ( s ) - .)0.000622s + 0.0151

(;8 ( s ) -

0.2795(is'+4.4(_5()_+16.(,TT I

(_.000622s + 0.0t50.)

0.27950s-- t. t6S(/s_lO.oTT

438

M 10 -Z _=.-a

gn

i 18-31 ....t --u

dc

le-4 _

1_-5 -2L •

le -6 _

le -7 i I i I iilj

10 -3 le -2

Gl (OFt Ctrl with F/B _ F/F, St}stem Xnovn Exact

\

\\

\\;

J

\\

\

45

0

-45

\

\

le 2 lO 3

Frequenc 9 (rad/sec)

-1 tOO le

Ph4

$

@

Figure 3

439

fl I0 -Za

gn

it I0-31

10 -4 _.

16 -5 :_

10 -6 =

101(_-3

GZ (Opt Ctrl vith F/B Onl_l, _Istem Known Exactl_l)

i u JJlLl _ , , i,,ll

10-1 leo

\

\

\

\

\

\

\

i i | ,llll

10

\\

\

\

45l:c

e

-45

-90

-135

................ 180

10 Z 19 3

Frequency (rad/sec)

Figure 4

44()

rl 10 -2a

g11

1£ 10 -3

U

d

10 -4

10 -5 =

10 -6 __-

10 -7

I0 -3

C_ (Opt Ctr) vitb "/B & F/F0 5 stem $tifrness Estimate Poor)

\

\

\

\

\\

\

\\

\

'\

\

"_ \\"\ \

//

\

/

/

/

/

\

ecL_

-45

-90

-135

-180

................. ZZ5

10 Z 10 3

Frequencg (rad/sec)

FiKure ,5

441

n t9 -Z41

gn

it 16-3

U

d

10 -q

10 -5

lO-b

10-7 , .......

10-3 10 -Z

Ctrl uith W/l OnlM, _ stem Stiffness

\

\

\

\

\

| . i l,l|* i _ 1 II,*J

10- l tO 0

£st lmmte Poor)

\\

\\

\

\ \ /\ /

\

\-

, _ ii,|*lt J t i LllJ

10 1 10 Z

Frequenc 9 (tad/see)

0

-45

-90

-135

-18o

........ -Z25

le 3

Ph&

S

e

Figure 6

442

Ctrl Nitbn9 Estimate Poor)

\

le Z le 3

Frequencg (rad/sec)

Figure 7

443

IS le -za

gM

1t le-3

U

d

le -4

le-5

1@-0

G6 Ctrl with FIB Onl_ Esti_te Poor)

\

\%

9OL_

-9O

'-180

-Z_

Phd

S

C

i i _ ,,l| , , i i,,Jl --360

10 Z 10 3

Fr_luem= 9 (rad/s_c)

Figure 8

444

M 16 -Z =&

gn

! -3t 10 __,,U

d

10-4 _

10-5 :

18-6 _

G? (Opt Ctrl with F/B a F/F; S_lstem k,c,m Estimates Poor)

\\

\

\

\.\

0L_

-45

-7I0 ........

10 -3 19-Z

L i , iJIJl!

-135

-180

, i Lllll , _ l,I*_, I , llJ_LA J L . ILiJJ --_-_5

19 -1 lO 6 101 11)2 193

Frequency (tad/see)

Pb4

S

C

Figure !)

445

N 10 -Z :41

gn

it 10-3:

U

dC

I0 -4 =

10 .5

10 .6 :

10 -?10 -3

GB (_ ,t Ctrl with F/B Only; _rstms k,c,m Estimates Poor)

\

\ \

\ I

L. ' L J,,,, | , i l,,,J i _ . ..J,i i , , sit.,

-Z le - ! lee le le z le 3

Frequency (rad/sec)

0L_

-45

-90J

t

f

_.. -135

\\

-180

........ -ZZ5

Pha

s

Figure 10

446

_ University of Virginia ROMAC

Experiment Compartment

Umbilical Connection

Scientific Experiment in Spacecraft

J447


DISTURBANCE LEVELS

Quasi-Steady or "DC" Accelerations

Relative Gravity Freauencv (Hz) Sourqe

1E-7 0 to 1E-3 Aerodynamic Drag

1E-8 0 to 1E-3 Light Pressure

IE-7 0 to IE-3 Gra vity Gradient

Periodic Accelerations

Relative Gravity Freauency (Hz)

2E-2 9

2E-3 5 to 20

Sourqe

Thruster Fire

(orbital)

Crew Motion

2E-4 17 Ku Band Antenna

Non-Periodic Accelerations

Relative Gravity

1E-4

1E-4

Freouency (Hz)

1

1

Source

Thruster Fire(Attitudinal)

Crew Push-Off

44,_


_enerolII I I

Probl em ,

In _he upcoming

s_a_ion, planned

spaceNASA

for comple _ion in _-he I?_)0 'S_

accelera s _rans m,_t: ecl1 i I I II I I " -

_rom _he space s_a ÷ i on _o an• _p • ri men _a I pIa_fo rm conga;ned

on (inside) the space $tation.

(¢ . ,,minimize : .reduce _-o -_ 10 .6,f possible

"low _Crt_ucncy": o. oo! _o _0 HZ

4 4 9

_ University of Virginia ROMAC\

Lorcntz

Actuator

MagneticSupport

Experiment MagneticMass

Support

Shaker

Concrete Base

Control Law Validation Apparatus

450

_ University of VirginiaROMAC

X X

X X X

X X

X X

X X_X

1

X X X X

K X X X X

F

K X X X X

X X X X X X XX X X

X

X

X

X

The Lorentz Equation: F = il X B

F = Force " 1 = length of wire i = current

X Represents the tail feathers of a magnetic field B vector into the page

J45]


\Sh.tkcr

Lorentz Actuator

DesignSection A-A

Large

Air Gap

,dllllllF_."_llll_coi,l_rCoil --.--411111E_Jr_11_'_- _IIIIIIL_ Magnet

Rod Connecdng

Iron

to Ex_nment

| "_:_=,._[_i__i_h___l_{_-_

MlftllIHI_! _Z_II1lIIilI1lliIlljIjI11liiii_IllI_lIl11IiII1l___F-

Permanent Magnet InsulatingLargeAir Gap

Connecting Magnet Iron PlalePlate

Cross-scclional View of Aclualt)r

\452


tJ/i

//]/

]

J/

C on'irollii,"

/]//N/I/I/ IlII/IIIlIlI/I

i"l"l

d(t)

Sys'tem Model

453


Objec÷ive:

((Fi nd "('he be s_"ee

minim; ze x ('(').

454


\

E Clua 4:ion o4: _ o_ion :IIII I I |II III

•. -. I]x ('_)= m'

\455


_S_a'['e Ec_ua'_ions:

ORB

x=Ax+bu+_ cmB n _

x2 C¢)

m

{ o }_o __

456


ProblemILl I

S_a_emen _ :I I

"Tr?

Determ;ne ,he control u(_')

wh;ch m;n;m;zes _he performance

indev --- --- -- - - - - - -- ----7

m I w g

concl;_ ;on $

I_or

t$u jec_ _o '(:he

' x o)= xI - --0II l im _x('_} = £I t: .--_oeI

I;m _c(,)= @

11

i

I!IiI

11I

11

11

457

_ University of Virginia

fS o l.,io_ M • _ho_l

ROMAC

( D i "F_e. ren_- ia I

E_ua_ions Approach) :

I. Au_men_ _he performance ;nde. x _T

Lagran_e m-l÷; pliers.

2. Take _he I s_ rattan;on _ _ o_ ±he

•au_men_e.d per_:ormanc.e Jndex and

Se_c ;± e_ual "to zero:

1

_e

_he ar b;_r ory vor;otl onS g X clncl

5.bs_;_u_ for _ in ÷ke st_e

to y ;e.ld

/_ BW;" B'r]f t-[w, +(:+}458

_ University of VirginiaROMAC

x (o)-- x.I;m x(*)= o

•_ ..+, oo

l;m T (,) : o

459


Solv, fo_ __(_);. _¢rms of

Such _:_a_ _:h_. rtma;n;n9 n

¢ons_cln_$ are _/;m;na_e_.

_. Use the t_u=_ ;on

"_o 1:;rid u* ;n terms o'F x.

X_ (_:} I_ a mannt

arbl'trary

Result:

: B ×=,X,, x

,,,..<-,,÷ -zz _2,f(*)a_:

! !̂]where iS "(:he .TordeIB

0 -

a._o., ca l Form of t_ Hamil+o.;+,

ma+rix I_'W, BW_"B'J,__,.

white h c.or,ia;ni only +he ne:_.+;ve

elcjenvalue.s of +he Hom;Itonlanw_cl_:r;X, correspondlnq _o the

llgellviill,le/ ot _" _he _lJoi_d-ioop

,y,,-.,, Co,,,.,,,;.,,j iA, Bl<..,,.oli,bl,J,

_h.,. X:[X,X,,X,]x,,;' +_'" .i,..,,.o+o,m-*,-ix which le.a, "to the

alDov+ .T. C, F, )

460


|0,

No_e:

w_ere ){ "'= -".y,(o) ( _ )"'tl X11

a._ where

I.*¢Sr-*im 5

|'n_eqra_;on e_ the ;nde_C,'n;_e

;_esral rt_;res constants

of ,_e:_ra_; on _-ha_ ea re e []

ide_;ce/ly zero.

X,.,,,^,_,._.T,x, ^ x"= 7,,, A- ew;'e'Pi11, .... 11 II

deve]op

P,,-X2sX _' [Pis _h,_ so(ution "('o "thewLII- known _(gtbra_c

R;cca't_ I_qua_ion] ;

aqu;v_len_ _ormS _or U ci'('t_:

_.u_*(.e)= - W_"B_"Px + W_'e" ,,,,="_'""e'^")er^_._,,"_"_f(,}a,

= - W._' B=P x - W_"BT ..==:Y_'_",._o(-A-)'"*'X_,_f_'') _''

w;' - -"s*=_¢ _ran.;_c;on ma_r;x approach y;elds

..u*(._)= - W_" B'P x_- W_" B" Y¢'"'_"^+..,_ ,[ e^t"_"_Tx_,__ a=

461


So[u÷ion,L

_u*("_ : -W,;' B"P x + W; "'B'''c''_'''z2e'^* ,1"• ^* '_z,"("_-?(,) a-t"

w;'8"P,, w;' B""""" -^"f _"""""

: -w;',"P_- w;'B"x,,"°"__(-^-')"" x'"'__'"rmO ZI :

: - w;'B"P,_- w;'B"_ (-_,-')"*'Pf("_r'=o

W-,B_'Px W3B'r -"" ,)"*' ",'_f..-):- _- -' x,,,.zo(-^- x,,

D,.opl,;,, _ 6;gh.r oraer e.r.,s (_o_ r-o):

¢o)_.'.It.(.._) tlt-lnT_ i¢,.-I nT_l(-l)'l_-I _!("1_ ,.-w_ _ rx_ +vv s o 1,zz /_ _¢, S

-. ,11--1 aTn till "1 D T V ('1)'1 A -I V ¢.1} P '_- V 3 O r _X + vv3 o ^zz _' "zz ""

-I T -m T -'r -'r T-W_ B Px 4- Wa B X,, A X.P-¢_

-- -I T -I 1' -T -1" TW_ B P __÷ W3 e X,, h X=,__

wh,re _.T= -P(Ph ÷.W,)"= (_-BW3'BTp) "T

Lx'-',,^ x';' '_ " "'"" "" ""''_= ) " _'zz i_ ^zz

These.are. several "forms for "the co.trol law.

J462

_., University of Virginia ROMAC

\

Solu't'i on:I I!

(-w;'B'P)+(w;' p)

_ Univer$ity of Virginia ROMAC

A Jica_ion _'o our

specific problem :i I

[ (.e) Cpx ¢_)÷ Cvk _,)

+ Ca a¢,) • Ca a (_)0 I

where Cp_C _, CcloDand Ca,

are con $_'a n _" _clinS.

\4fi4


S_ D (s_

F-

IJI

D C.s) ._

It

I(

J

Block Di,,5_m

465

University of Virginia ROMAC

\

Table I Optimal F/F and F/B Gains for Selected StateVariable and ontrol Weightings.

System Parameters:m = I00 Ibm k = 0.3 lbf/ftc = 0.000622 lbf-sec/ft (( = 0.1%)a = 10 lbf/amp

_, igh I,:',

_la[ _lh

---4-

2 I

I0 "2

23 I 3

41 4

64 5

92 6

12ti 7

165 8

2{}cI !J

25,'_ IO

5_1 15

IO3,1 20

I(i17 25

2:12q 30

;| I TI ;|5

,I 1I:1 II)

!I:1:15 60

165141 t_O

25911 I00

--v_

w:I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

t

I

I

F/IfGa iu,_

(', Cp v

I. 3845 i. 3637

3. 1324 !. 9863

4.7659 2.4413

6.3732 2.8210

7.9701 3. 1544

9.5617 3.4552

Il.1950 3.7354

12._115:| 3.9919

14 .,1'21;'_ .I. 23_0

16.032.1 ,l.1674

24. 0740 5..1729

32.12S9 6, ;120!!

I0. IKI9 7.0680

,1_.2297 7,7,1:11

56.2_1(i t_, 36,1(i

Irl .:1:¢61 _.9420

96.5:lfJfl IO. 9.r_26

12g.7372 12.6475

160.9389 14. 1407

("dO Cdl

0.0294 -0.0006

O. 0297 -0.0001

O. 0298 -0.0000

O. 0299 O. 0000

O. 0299 O. 00O0

0.02_J O. 0000

O.0299 O. 0000

It. 0299 0. 0000

O, 0299 0.O0_X)

O. O299 O.0000

O. O3OO O. 00O I

O.0;100 O. 000 1

t) 03011 0.0001

0.0300 0.0001

0,(;300 0.0001

O. 03(X) O. 000 l

O, 0300 O. 0001

0,0:100 O. Of)OI

O. 0'300 O. O(.HJ 1

I-'II"(;ai,s

Cd2 Cd3

4). 0067-0.0070

-0.003O

41.002O

-0.0015

-0.0012

-0.0019

-0.0010

4). 0007

-0.0005

4). O0O9

4). 0004

4). 0002

-0.000 i

CdI Cd5

4). 00,19 4). 0032

4). 0004

-0. 0001

-0.0001

-0.0000

-4). 0010

-0.0008

41.0007

-0. OOO6

-0. 0006

-0. OO04

4). iX)03

4). iX)02

41,0002

41,0002

4J. [KI(Jl

4}, o0o I

-4),o0o l

4). 0OO I

4). 0004

-0.0003

4). 0002

4). 0O02

4). 0002

4).0001

4), 000 I

4). 0000

4). 000O

4). o000

4}.(}(M_,J

-0. o(,_0

4),0000

4). 0000

4). Or)' )1 -0. 0000

-0.00( t -0,0000

4). 0001 41. 0000

-4). OffOO -4). 0o00

-4).0000 4).000o

-0. 0000 4). 00(_}

-0. 0000

4). Of)O0

4J. (l(_()0

-0.0_00

4). O0 }0

-o, f)ooo

-0. 0000

-4) 0000

................... i

)

-0. o<XX)

4). 0090

-0. (x_)o

-0,0000

4). OfX}o

4). 0000

-0.0000

-0. _X)O

466

c +,


Table 2. Closed loop transfer functions for system with designparameter values of k = 0.3, c = 0.000622, and m = 100; butwith actual parameter values as shown. 61, G3, 65, and 67include both LqR F/B and proportional F/F; G2, G4, 66, and G8include LqR F/B alone. Weighting parameters used were

Wta = 258, Wlb = lO, w3 = 1 (see Table i).

System Parameters

, Ibf_k t tirC-_

0.3

0.3

0.45

c (Ibftsec) m (Ibm)

0.000622 100

(/:o.1%)

0.000622 100

,e . ,

0.000622 100

0.45 0.0006'22 100

al(s) =

Closed Loop Transfer Function

s2X(s)s2D(s)

0.0000622s + 0.0001

0.31056s2+4.4675s+i6 0524

G2(s) -

I

G3(s) -

O. :3 O. 00622 100

0.3 0.00622 100

0.45 ).00622 90

0.45 ().00622

G4(s) -

0.0000622s + 0.0300

0.31056s2+4.4675s÷16 0624

0 0000622s ÷ 0.0151

0.31056s2÷4.4675s÷16 0774

0 0000622s , 0.0450

O.:31056s2,4.4675s+16 0774

0 000622s , 0.0()01Gb(s) -

0.31056s2÷4.4680s+t6 0624

0 000622s ÷ 0.0300G6(s) -

0.31056s2+4.46_0s+16 062_ i_, , , ,, I

G7(s) 0 000622s _- 0.0!51.-)

0.27950s'-,4.46SOs*i6 077

90 _8(s) -0 000622s * ().0450

.3

0.27950s'-4.46SOs÷16.077_

J

O_IG!N._L PAGE IS

OF POOR Q4JALITY

467


M i0 -z(I

gn

1t 10-3 :

U

d@

I0 -4 _=

10 -5 =

le -6 =

10 -?10-3

Gl (Opt Ctri ulth F/B 41 Ir/]F, S_stem Knmm I_xactll )w

f

\

\\

\

\

j_

i i 1 JJ|,_, J a m ll_Jl

lO -1 10 0 19 1

\

\

\

cL_

\

\

,\

\

45

-45

-99

-135

........ 186

10Z 103

Frequen_ (tad/see)

Pha

s

e

Figure 3

J468


M le -z

gn

it 19-3

U

d

19-5 _

19-6 _..

I0-710-3

GZ (Opt,.Ctri

i l , , i1,, i l i i i1,1

19 -Z 10 - 1

with FIB Only, _Jstem i{nown £xacth

\

\

\

\

k

\

\

\

lcc

lo I

'15

-45

-9O

-135

\\

\

\

I0Z le3

Frequency (red/see)

Pha

S

e

Figure 4

\469


II le -Z8

gn

lt le-3

U

d

le -4

le -5 =

le-6 _:

le -?10 -3

_3 (opt Ctrl with F/B S F/F, _ten Stiffness Estl_te Poor)

'\\,

\\

\

\

\

ecL_

-45

-90

\

_,. X -135

-18e

.................. ZZ5

10Z 10 3

Frequent V (rad/sec)

\

\

\

/

#

/

/

/

/

\

\

x

\

, _lll , l , ,lit| i , llllJA

10 - 1 190 le

Ph8

S

C

Figure 5

)470


M 10 -z =A

gn

1t 10-3 --U

d

le-4 _

10 -5 _

10-6' ..

10 .7-3

IO

64 (Opt Ctrl with

* i Jl ,i*Jl . • [ JILL

10 -2 10

ir/B Onl_l, _ stem Stiffness £stim_te Poor)cL_

\

\

\

\

/

\

\

.... 45

\

, \\

\\ i

\ /

\

19 1 10Z lO 3

Fr_-_luency(ead/sec)

Pha

s

f_

Figure 6

471


M le -za

gII

it le-3

U

d@

le-I

le-'5

le-6

G5 (Opt Ctrl with F/B a F/F,

-3, J A ,Jl,[ l J i J lli_

I0 -Z 10-1

I

I

I

./J

J

S _tem Danp

'\

\

P

jl/,

10

imj Est lut,

\

\\

Poor )

\

, , L lJ_J

le Z

Frequencg (rad/sec)

9e

e

-90

-1Be

-Z'_

16 3

Figure 7

\472


II le-z =A

gn

|t le-3 -

tl

d

le -4 _

le -5 _

to -6 =

le -?le -3

G6 (_

l i i i_., i • _ _*Jll

IB -Z 19

,t Ctrl with F/B Only. _stem Danplng Estimate Poor)

\

\

\

i _ _ [li._ i . ll,,l_ l . , ,.ll.

1 100 10 1 102 103

Frequenc g (rad/sec)

96cL_

0

-186

-Z?e

Ph4

S

FigHre 8

473


el le -Z =

gn

tt le-3

U

dC

19-I i

le -5 =

le -6 =

7le-7

I0-3

G"/ (I]

19 -Z

)t Ctrl with F/B A F/F; 5_stem k,c,m Estimates Poor)

\

\

\

\\

"\\

\

, J i _,,A|

101

0l:c

-45

L

"_ -135

\,

-180

.......-ZZ5

10 z 10 3

Frequencg (radlseC)

Fi_,ure 9

474


II le -z =-&

gn

1t 10-3 -

U

de

le -4 =.

le-5 :

r-

19-6

10-710-3

68 ,t Ctrl with F/B 0.11 k.c.n Estimates Poor)

"\

\\

\

\

\

cL_

\\

"\

\

\ / f

",,. /

"b

I

\

\

s i z i,[,l , i , , ill,

- 1 100 le I

-45

-9O

-135

- 180

...... -ZZ5l L i i,_J

lO 2 lO 3

Frequency (rad/sec)

Pha

s

Fi_zre tO

47,5


C onclu sions,[ I li

II

4

e

A n op't"imal con't"rol kin, ID..en+or

An =pproxl'ma_;on +o +hl's

has been found which roses

de_c erm ine_l

+he n°nhomosentou, LQ R problem.

OF* i m=l ¢on+rol

"FeLal belck

Th_ op't";m= Iadvan+ag_s :

and _eeal _or wclral_alns.

con+rol has +he _:+llow;n_

• I

=. The :_n,.s c=n be eal;ly ele_'ermlne_.

b. The, control ;s very robust ((;O=ph=seB e 0

malr:_lnj ;n_;ni+e posi+;ve _oln vnar_an)

• . )C. "rht con+rol is =pplicabl_ _o = wide

ran:_e of' problems.

el. Th= con+ro[ o_c_ers subs+an++=l

;mprovemen_" In dlS_:Ur_elnce rejection

ove.r _=_C _l++orell_.d by LQ R Fe¢=[loa_k

0

e. _¢ =o._roi can be e=s;Iy Impl_-mP.n_¢cJ.

476

n91-, 1206 - nasa · means is needed to preveul the experimem from driftin,,_ inlo ils own...

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