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Design of Reliable Control Systems with Guaranteed Disturbance Rejection Performance
Copyright © IFAC Design Methods of Control Systems. Zuri ch. Switzerland. 1991
DESIGN OF RELIABLE CONTROL SYSTEMS WITH GUARANTEED DISTURBANCE REJECTION PERFORMANCE J. V. l\1edaniC*, W. R. Perkins*t, and R. J. Veillette** *Coordinaled Science Laboratory. Unive rsity of Illin ois. 110 1 Wesl Sp ringfield Avenue. Urbana. 1L 61801. USA **Deparcmenl of Eleclrical Engineering. Uni versity of Akron. Akrvn. OH 44315-3904. USA
A bst ract. A met hodolog\' is presented for the design of control systems that remain stab le in the presence of a pre,pec ified subset of possible actuator and sensor failures . and that maintain a prescribed le\'el of performance (distlll'bance rejection) as measured by an H ",,-norm bound. Both central ized and decen trali zed control:, rtlctllJ'(~S can be treat ed. Thi s ne\\' methodology is based on a modification of the Ric cali approach 10 " ",- norm optimization. An cxam ple illustrates the procedure. E ey \\·ord s.
relia bility. robu st control. se nsors and actua tors. large scale systems where = is the controlled output. and .'I is the measured outpu 1. and cons ider the co nt rolle r
Introduction COl1\'entional f,,('dbilck conl rol d"signs for a mul ti-inp ut. multi-oulput plant may reslllt in IlI}Sal isfiIClory control system p('J'forman('('. or e\'l:n instahility. in the (,1'('111 of actuato r or se nso r outng('S. ;\ COIII rnl S~'SICIII d('sigllf'd to lolerale failures of actuators or of 5('n,ors \\'ithill a prcspecified subse t of all pos sible fai lures. \\'llik retilining de, irrci contro l s ~'slell1 properties. \\'ill be called a "r('li"bl,," conlrul sy,;tcm. This paper considers ll1('t hodologica I 1"'O(,l'rill n's for I hp design of reliable contro l sys tem s.
, (
u
z
= (H.T), U
Y= Cx+
Let the
where
Fe
A Ell ] [ LC A+G'lld+EI(-LC '
Then a key lemma on bounded real systems states that the resulting system is stable and satisfies
if ( Fe. He) is detectable and the algebraic Riccati inequality
has a solution Xe :::: O. Thus. an IJx-norm-bounding control will result if J(. I\d. and L are chosen so th at these conditions a re satisfied. The approach used for deriving reliable designs in [8,11] to express the Riccati inequality sufficient condition as
F; Xe + XeFe
+ ~XeC.C; X. + H; He + p.
= 0,
IS
(1.2 )
Q
where Pe :::: 0 is ullSpeci fied. To ach ieve rel iability, we pick p. such that the closed-loop system satisfies the key lemma not on ly when all control components are operating properly, but also in contingency situations when some sensors or actuators have failed. Cons ider a design that must tolerate particular sensor outages. Perturbations 6J.F. and 6J.H. are identified which correspond to fail ures of the susceptible sensors , so that the triple (Fe + 6J.F., C" H. + 6J.He) then defines the plant in a n outage situation. By adding and subt racting
(l.la) Wm
CO
.r, = F,.T, + Cew.
To introduce the approach consider the system
= kc + Bu + CWo
+ Bu + Ct;:o + L(y -
where t(·o is a model of the process di sturbance. resulting closecl-Ioop system be
There have been relati\'el:- fell' pre\'iolls attempts to develop methodologies for the design of reliable contro l system s. and these have had various reliability goals [12.2 ,1,6,7.9.3]. These designs hal'c not how('I'{'I' . addrl"ssed the isslle of pro\'icling guarantees on system pcrforma.nce. Thf' incorpora tion of performance guarantees \·ia a bound on the H"" norm, in addition to closed -loop stab ility. is a crucial contribution of the approach initiiltpd in [8.1 J]. Designs proposed in [11] arc characterized by closed-loop stabilit\· and H x norm bound when all control component.s are operating. as \\'ell as for thp case when any admissibl" control-component fail'lres occur. In the centralized case. admissible failures co",isted of any actuator or sensor outages within a predefined se t of susceptible actualors or sensors: in thl' decentral ized ca'e. admi ss ible failures co nsisted o f any controller outages within a predefined set of susceptible co ntrol channels. The C'xistence of appropriate solutions to the developed design equa l ions is sufficient to guarantee the rel iable stabi lity and performance of the closed-loop system . The reliable designs el'oh'ed from the basic H"" designs given in [4] for the cent ra li zed case and in [10] for the decentralized case. In this paper \\'1" will consider additional types of sensor/actuator outages. and the loss of actuators and of state \'ariable measurements in the case of state-feedback controllers.
.i:
.-\ ~ },' ~
(1.1b)
tThis work was supported by th e Joint Services Electronics Program under Grant N00014-90-J-1270 , and Sundstrand Corporation.
239
appropriate terms. t he co ndition (I.?) is th en r('written in th e form
( F + ~F,)Ty, + X ,( F, + ~F,) +-;!-r X ,(;,(;; X , + ( He + ~H, )T(H, + ~H, ) = -P, + ~ F,TX, + .\, ~F. + H ;r ~H , -L ~I/ T 1/. -'- ~H.T ~1I .
Remarks. 1. \\'ith al l sen sors operatio nal. which corresponds to w' = 0. T:Js) = T(s) is the nominal closed-loop transferfUllction matrix from n' to z. where
1\.:3 )
.-\ specific P, ~ 0 is then drtrrlllinNI ,u,h that tllf' right-hand , id,' of (I .:l ) is negatiH' "·In i-ddinik. T helt . 11 .:\ ) imp lie,;
(F
+ ~F,)T X , + X , I F. + ~r ;)
+:f,X, G, G; X,
+ ( 11, + ~II , )1(1/ , + ~H ,) :S
O.
Theorem 2.1 cOl'ers this ca se automatically. since ~. = 0<;;; n. If senSOrS corresponding to a nonempty s ubset ~. ;: 11 fail. then T~(s) is the res ultin g transfer-function matrix from tc w to z . where
1 l.! I
so that tile perturbed S.I·'tC'm ""ti,fil'" t11t~ prillcipa l hypoth C's is of the ke.l· Iemllla. Ik"iglt "'1 l1atio ll " d('I'eloped ill :~ .ll l us in g thi:::i app roac h fu r th e <:il~(' ut' ~(' II!"or olltag{ ':-\ alld fo r
U' -. =
....
t he case of act ua to [' :, (,)u tag(':-; arc ~lI l 11l 11 a rized in t h(' 1\(':\ t
U'o ) (
.
tc m .:·
with IC", -, co ntaining only those components of mea surement noi se assoc iated with operational sensors. Tl lm . fo r the re liability formulati on in [11]. a se nso r failure effect il'ely e lim inates the associa ted se nso r noise. Th e p roblem whe re th e sen so r noi se pers ists although th e plant meas uremen t has b een lost will be treate d in Section :l.
sect ion. The Design Equatio ns for t hf' l3 a, i, :;('n,or and .\ ctllato r Fa il url' \I od,", Sensor outages .)
n <;;;
{I. ? .... dim l lil} corre'IJ()ltri {() a ,e!cn"d ",Il'(,t of sc nsor, s usc(' ptihl, ' to ollta~es . Introd uce the d"l'O lliposition ( ' = ( \2 + ( 'u . 12.1 ) 1. ('1
1l1 <-tt rix
f-1+ =
l ",
formed from C by zf'roing out rOIl's corresponding to SllS(,(,Ptible se nsors. 1.(' 1 .... S; 11 d('not(' s('n"ors that aCIII"lIy eXI)('ri e nce an outag('. and let TJs) den ot(' the trall sfe r-fun cti o n matrix of tll(' r('s lIiting clos('d- Ioop 51·, t(,l11. It is COItI"'lIi('lIt to adopt th e notation
=0 2
(2.:3)
CC + L",C", .
(:Z . I )
C~0Cn
0) > 0 - 0 '
~o
12:1)
+ I. ~.
(
Pl'OoI In lif'II' of Remark (:3) abol·e. if (2 ..5) and (2.6) have appropria t I' sol "t ion s. [-11 (2.8)
sat isfies
lI'here C", and C_. han: Illeanings ana logo ll's to tho s(' of Cn a nd Cn in (2.1 ). Since w' S; n. C~ C'" :S C~ Cn . Also d('colllpose the obsen-er gain a s
L = L~
C~J.
Th is identifies the matrix P, in (2 .2 ) as
wh e re Ct ! dC!l o l,'S Illl' ItIl'ihUr<'!lll'1ll Illat rix i1s."ociatcd wit h 11. a nd CIl denotes the l11ea SUI'I'llll'nl matrix "",o,iated witll t he complemen lary su bs(' t of m('aSllr('nwnt s. Till". Co is
c=c+c.
Th l' design equations (2. cj) and (2.6) can be I'ie wed as ari,ing frolll th e repla cement of I-I by th e aug me nt ed
(2.9 ) with }' = (;!, Xl + 02X) . .\10reol·er. it can a lso be show n that ( F, . !-I<+) is a d etecta ble pair , wh e re
so that
LC =
(2.10)
(That is . L: ha s zero co lum ll s cOlTl'sp()lldin~ to ,"mors I"hiell hal'e actua ll y failed.) TI1<'n thl' followin g r(,su lt Il old s:
Thi s guarant('es stabil it y an d the !-I x· norm bound 0 for the ca se no se nso r out ages when the syst e m is described by the matrices ( F, . G, . He). For se nsor outages corresponding to w' <;;; 11. t he controller dynam ic structure is not affected by a se nso r outage: only the controller input structure is changed . and the controlle r bf'co m es
Theorem 2 .1. A ssume ( A. f-1 ) is df'tpcta/Jle. ilnd/et X ~ 0 and }' > 0 satis(1' th e .·l /gf'iJrai, Riccilti Equations (.-IRE,)
ATX
AY
+ XA
- XBBTX +;!,XGCX + HT l/
+ YA T + },YH T H\' 0"
+ 02ClCn = O.
- \ 'clCn )'
+ GG T = O.
(2.'»
(2.6 )
Gil'e n 12 .11). the closed· loop sys tem matrices become
respectil-'ely. and the side condition
_ BT.\.- . X d = -;!-r GTX. (J - o2Xn-l} ·CT .
(2.12 )
(2.1)
G,,, =
lI'ith A + (;1\d + Bh' Hurll-itz. Then. (or sensor outages corresponding to any w' <;;; 0.. the closed-loop s.l-stem is sta ble. and IIT;;;II"" :S o.
from which it follow s that
(2.13a)
240
be (2.1:3) in (2.9) and the fact
F/~ ..Y, -i- S , F~ " -'-
CAC n 2: C!C
(2.136)
A ssume t he controller is open -loop (in rem a lly ) stable. Then , for act u ~ t or outages corresponding to any w' <::: 0 . the closedloop system is stable. and Il f"", ,, ::; o.
(2.J:3c)
The proof is omitted due to s pace limitations. It proceeds parallel to the proof of the pre\iou s theorems: see [11) .
lead5 to
R em ark s.
7X,r; ,J;;CX" + f(I fI , =
-c,r..J:_Y, - SJ ,J',_
1. For actuator outages corresponding to ~. <::: fl . T~(s) is the transfer-function matrix from (e to c:;,. where z:;, in cludes only those control components associated wit h operational actuators. Thus. the control signa ls asso· ci~ted with failed actu~tors are dropped from the regulated output c. and do not enter into the computat ion of the H x norm. The formulatio ns where these s ignals are retained in :. a lthough tht' can t rol loop containing the actuators are open. is consider(·d in Section-l .
-?\J ,J;~ X, - ()2 (cl) (C n 0 ) ::; -C}~L;_S, - S,L . f ,_ -?Y.L u L;".\", - ()2C/_C_ = - (~X, L , " + CleL) u:e~.Y, + oc,_) ::; o.
Hence. if (F,~. H , ) is a detectahle pair. then F, " is ]-I\lJ'w itz. and i :. \ <, ) = II ,(, I - r ;:,)- I(;.:_. the trall sfcr·fllllc t ion Inatri~ fronl IC" to c. satisfies IIT:!I" ::; (). The dc-tel"lability proof is routinc: If (.1 = (e? l'j) i 0 sat islic,; I ;J = ,\t· and flet· = O. then Ael = .\t·) alld 11 Cl = O. with (.-\. 11 ) assumed Cl detectable pair. Therefore . ('ilher F! c(.\) < 0 or Cl = O. If VI = O. then }< ~e = I ~~e = ,\e and 11 , ," = 0 gi\·cs H,+ ," = O. Since (F~ . He+) is a detectable pair. /i t(.\) < 0 Q.E.D.
2. T he design equations (2 .16) and (2.17) call be \·iewed as arising from the replacenwnl of C \\·ith the augmented rnatri~ G+ gin'n by (:2.1~). Thi s corresponds to choos-
.lflg P, =.\" , ( Bo0B~
0) 0 .\" , 2: 0 .
Actuator outages Sensor Outage '.l ode with Res idua l '\ lcasurement \"oise
Let 0 <::: {l. 2 ..... dim(u)} co rrespond to a selected su bse t of actuators susceptible· to outages. Introduce the decompo-
Consider again the sensor outage problem. but assume that the output chanllPls are always acti\·e and when a meas ureme nt is los t the sensor still injecls measurement noise into the sys tem. Thus. in a failure lllode
sition
B = Ho
+ l3 0 ·
(1. J.J )
where Bn denotes the control mat rix a ssoc iat ed with the set 0. and Bn denotes th e control nIatri~ ao;sociat('ci with the complemcntary sllbsct of control inputs. Tlltts. R {j is formed from B . by ze roing out columns corrcsponding to su,;,"cptible actuators. Let ~. <::: n correspond to actllators t hal a ctually fail. and let T:,.($) denot(' Ill(' transfer-fllnction rllat ri~ of the resu lting closed-loop sys tem. It is con\·cnient 10 ~dopt the notation (2.15 ) B = 13.., + 8.., .
y, y,
Ci,l"! IC,.
+
ll'i-
i
E~·
i ~ . .;.,' <::: rl.
(3.1 )
\\·het·" the mea surerne nt noi se yector is decomposed as U'~ = [U·I (CZ .·. IC, ) . The controller in a failure mode is t.hen
where B~ and B ~ ha\"(' meanings analogou s to those of 130 and Ro in ( 2.1~). S ince~· <::: n. RJJ! ::; /J,, 13~·. c\ lso . decompose the feedback gain as
U
}\.(
~
(A + BX - LC + GJ\"d)( + !. ",y +
L~w m
(3 .2)
so t hilt t he closed loop system is characterized by
so that
. (GO) 0 L .
The o rem 2. 2. Let .\. 2: 0 and l - > 0 sat isf\ the AREs
AY
••T
+CC
+ 0 Z E oBoT
=
(HO) 0 I{
.
(2.17)
o.
respecti,·el.\ ·. and the side condition lT~axfXI ·) < define
He =
:\ote tha t retaining the measurement noise associated with failing sensors means that G, remains as in the base case. Transforming to the {1·1, d coordi nates produces now
+ y,.\T + J,YHTH Y - l -CTCT o
G, =
Cl'.
Also
(2.18b) and let the controller be
(2. 19a) u=
J{~.
(3.4)
(2.19b)
241
We again seek
Pe
in addition /\ d is defined as in (3. t:l ). Fin a lly (3 .10c) reduces to (3.12) when L is defined by (:3.1 -1) \I-ith 11 = ,,'XI-I.
so that
Th(" required detect ability condition on ( l-~ ~ , H ,) is established exact ly as in the case of Theorem :!.l.
Y,
p05Sesses a solut ion 2: O. Assuming a block diagonal solu tion as in (':' .8) and using (:1.-1) allol\"s (:3.,3) to be rearran~ed as
r/~ Y,
\01\"
+ Y, r~ ", -r- -f,Y, C~',(;; _\7, -i-1T[H, = -Pe - CfjLY, - _\:JuC~
It is not ed that the ."REs (:3.11). 1:3.1':' ) that ,wed to be so h'ed in sequenc(" can be replaced b,' two completely decoupled algebraic Riccati equiltiom'. Spe<·ifically. the _-\ HE (:3.1:2) can be replaced by _-t} -
+ }- ·11
-
if
I qC[) }- + f,lll'/f} + } CAC n }- ~ (;(;1 = 0
(31;j)
\I'ith \\- and}- related by 1-' = \1 '-',,,'.\: Ill<' reqllirelllent 11- > 0 in TI"'(""'111 :LI del11and, tililt tl\(' sid" condition ~2 > IT,~""I Xl ) lw illcllldf'd . .-\n .\ It<'rn,ll iw' .\('\IIi1tor-Olltil ,'!.I' \l"de
lL\;' + \ , /~c + f,-\ ,{;,{,-;5:, + Ji? 1-1e
s - (nC}_.j. +X,f_,~)(etC," + +r;~X,):
so. by prnioll> Mgllme'IIb. I he system I\"ill be SI able and haw' gllaran!(:'I'd 11,- -nurn, I> OllIH.l et for th e considered failure mode prO\'i c!cd (:l ..'») is sa ti sfied.
it
We now u se the slIfricicnt conditions (:3 ..5) and (:3. / ) to determine thp corr('sponding design equation s . First note that choosing as
P,
\"'hen all aclllator fails, we may still "'ish to de!!land that Ilw control e ,wrgy of the faikd ilClualor I", bounded e\'t'n though t he loop is brokell. To this effect, W(' cOIl., ider the output signal to be li; = 0_ i E ~. ~ as far a, th!;' system dyn.aJ11ics are cOllcflrn('d. but aSSllIlW that Ili = ]\·lE.. i E ..t-' as far a, the regulilted outputs are cuncellH'd.
n_
Fora giH'1l failure mode_ the cOlltrolle rtll('1I ha s th('structure (1.1 )
(:39) gllaranleps tl,al (:\./) ",ill hold for "ny f"ilurf's~' ~ O. ),h"lI. recalling tllf' block-diagonal st rncture of Y,. and parlit.ioning (:l ..~). \IT oblain Ih" Ihrf'c Ctl11dilions X(A
+ RJ,') + (A + + Il T/-{
Ell\ )' X
+
,+,XC;C;TX
+ 1\11,- + 02'C·0.Cn
= 0
with !\. !'-d+ ilnd L to he dc\r'\'IlIilwd. TIlf' clos"d-Ioop systelll is tllf'n characterized b\· Ill<' t ripkl
(-1.:2 ) (TI0o)
. ((;0) 0 [
('e =
+
X,(.-I - LC G /'-d) +bX,CCTX,
+ (A -
LC
+ GI'-d)TX,
+ -;}rX,UTX 1 + I,-IJ,+bX1InL'{;_\-1
.Ii, =
(110) 0 /,-
or after a coordinate transformation by the triplet
(:3 .101')
= 0_
A. suificipnt condit ion for this to hold is captured in the fol-
( 1.:1 )
lowing Theorem: Theorem 3,1. S'lppose (A. H ) is detcctabJf'. and Jet X 2: 0 sat is(\' the ARE ATX + XA - XBBTX +c!>- X GC;TX + HI H + o2C6'Cn = O.
and ll" (~1
(3.11 )
\011" assul11ing a (:3 ..'i) -type condition holds for the case of no failure. it then folloll"s that in a failure Illor\t'
R",( _\(, )
> 0 satisZ\' the .-iRE
+ GJ\d)W + IF (A + Gl\d)T -
II"C0.Cn l\'
+bli-1,-T IdF + GG T = o.
:=::
I~1;_\(, + Y, F~" +7X/;-,C';X, + 77;77, -P, - X,BuK,~ - K,r.i3L-\;',
(-1-1)
(3.12) Therefore. choosing
P,
to guarantee
\\-ith (:3.13) Then. for senso!' outages corresponding to any ",' ~ n. the closed Joop system, ,,-ith the controJJer and observer gains
is stable, and
I{ = _BTX. IIT",II."" < 0_
L
= WCT
\I-ill result in a reliable design. It \I'ill noli' be demonstrated that choosing
(3.14)
Proof. Equation (3.1 0a ) reduces to (3.12) when f{ is defined as in (3.14) . Condition (:3.10b) is identically satisfied when
242
leads to a computationally fea sible reliable design. This follows since , for P, as in (-1.6). equation (-1.-1) becomes
~ 0
lI'hich is a sufficient condition for 11 T.oII ", to hold.
a nd 11 [ 11 x
lI'ith '/2 # 0: t hat is. ,\ is an eigen \'alue of the ope n-loop controller dynamics . Thus . if assumption (i) holds. then Re(A) < O. Suppose (i) doE'S not hold.
~ 0
The design cCjuiltions can be dcr i" ed from (:3.,,)) and (~.6) . .-\SSUIllC agii in I to ha"c the block-diil!:>;ona l >trllct ll rc (:2.S). TheT! from (·1. 6 )
j'J,
SincE' (-I I:")) gi\es
I\-'/2
= O.
(~. IG )
gi" es (4.1 j' )
:'II ultiply ing (-I.9c) on the left by v;. and on the right by V2' ilnd llsing (-1.17) and the re lalion XIL = 02C 1'. lI'eobtain
=
(-I.IS )
If inequalit\· holds in (-I.IS). t hen Rtl .\) < 0: if ('f]uil lit\, holds. then C"2 = O. ilnd XI > 0 implies Rd.\) = O. To 1'1'('rill(k tl'is 1'\I,sil.,ilily. eith ... 1' as'tl111ption (i il 01' (i ii ) is surtici(,11!. sincc' (·1 1j' ) lI'ith ('''2 = 0 .Hld 1""2 = 0 gin's
and (:3.-5) can 1)(' partitioned into the threE' condilions .\ 1.-1 -;- J3J, ).l. ( .-1 + lJ J, )' .\ -;- /IT 11 -+- [,'1' [,' ..;.. .\' R" 13~.\
X fJ [,' -i.\' I
( ,-I
+ -;'.\(;C;T\ T
'.! [,',~ [,' f' =
n
J,'3~ C:T ' \ 1 - -"-: .W;(; F \ I - .\ :f3,:r-J?1.\'1 :.;. [('[,' = 0
+ ( ; + [," !-i-
- (. L ) -+- ( . 1 + (;~ [,'f L )J .\ I +7.\'1(;(;T\1 + -f;'\ILi.F\1 -+- J, ' [,' + XIH,:/J~X I + '1.1'·fi ['-n = n.
where (;+ is gi"eT! in
T
-
1 1.\),,1
(.1. ~ 1(, )
(.
(~ .9(·)
Consider no\\' t llf' statf'-rt'e dback con t rol problem of finding co ntrol lall' 11 = !\'.l' lI'ith tl1<' obj('cti\'(> to guarantee a prescribed fi x-no rm bound for th(' transfn-fu nction matrix from /I' to : = (~,"). d('spite any actuator outages lI'ithin a S('il'ct( 'd sul,s"t (l illl a\'ailabil' actuators, .-\ ss u1l1ing a state kec!back con t ro l. t he closed-loop syst<'n l becollles
(:2.I~).
it
T h eo r e m 4 .1. .. \ SS IIIll(, ( ·1.11) i, cietertable' ilnd Ie't .\ :::: 0 sat isf,' lite ..l, RF. . IT.\'
+ X.-I -.\' 1J(!Bi!'\
+:2 .\' IJnB~X
+~XC(;I X
+ N TH
= ()
i l.IO )
.r
I(s )
+
+~ H' 1,'f~ [,'0 I I ' C;(; 1 0' l3nlJ~ = U
+
(~,II
(A
)
(-1 .1'1. )
is st ablc. anci II T.A S)II x <
Cl
fnr \I'it h am'
(.·1 + J3 1, )Ty
-
. 1 + J3", 1\':;1x + (ffT 1,-J)
<:;; fl fai l. th" closed-loop
:
= (::).1'
A - B:..J'·.X 1 (;
+ X(. ·\ + B X ) + ~ X(; (;T X
m+
Pro o f. It is pasily sho\\'n that choosing X as in (,1.12 ) rpduces (-1.9a) to (-l.1 0) whence choosing 1{'1 + as in ( 1.21) identical ly sa ti sfies (-l.9b ). Finally choosing L as in (-1.1:1) \I'ith II' = 02X I- 1 . rf'Cluces coTiClition (·1.9c) to (·1.11 ) . It remains onll' to sh oll' thilt tll(' 5\'stem is detectable. To tlli s pnd. Irt I1 ~ (~:) # 0 sa t isfy ,
X (.-I
+ l3j,' N) + ~Xc;c;1'.y
= - p -
[,.! B~ x -
(.5. :3)
X B~ 1\·~.
(5 . ~
)
[,.!
If P + B~ x + X R~ X~. :::: O. then (.5.-1) is a sufficient condition for the system (.5.~) to satisf\' the H x -norm bound. Ch oosi ng P = I\'l I,'n + X Bn BTz X is sa t isfactory. and choosing /\' = -BTX . lead s to the design equation .-ITX
+ X A + ~ )( G(;T X
+ X B n Blx - XBnB'hX
Sholl'ing t ha t Rd A)
(:3. 1 )
whne (.-\ + 1]],'. (~)) mu st bp detectab le. and f' :::: O. From C'd). II'P obtain
<:;; fl
( ~.I -I
= (;:').1'
C;
+(11T 1, T)(;n + p = O.
( 1.1:3 ) ~.
se t~·
I
T Ilf' sufficient condition for (.'d) to b(> lI x -norm-bounding for Ih(' nominal ,ystCIll is t hat some X :::: 0 sat isfy
Then . if (i) I he COIII roller is st aiJlc or (ii ) .. I + (;+ /\'.1+ + Jj 11' i., I-I u}'\"itz 0' (iii) .-1 + (;+/"J+ has no j~··i1.'\i-, cigcll\,i1ll1c,'. I}",· closed- loop s.,'slem \I'iUI controller gain (4. 20) and o/)serl'(:" gain L = II C T
:
+ Uj,'",):r + (;(1',
(;~)(s [
\\'ith 1\- = -B'.\-.
+ J-J!\').1 + ( ; U'.
(:n (,[ - .\ - B [\- )-
If actuiltors assoc iatf'd with a syst(, lll bCCOllWS
(A + G+ 1,'d+) 11' + I I' ( A + (;+ 11',1+) + +,- 11'[,' '/ 11' 11'
_ II'('T(' II'
( I
+ fl T fi
= O.
(.5.5 )
)
T he detectability is easily \·e rified. anc! so the design guarantees the desi red reliable disturbance attenuation. The following thE'orem summarizes the result.
(1.1.5 )
Th e ore m 5 .1. Let X :::: 0 satis f\ ' (5.5). and let 1\ = _ B T); , .--I ssume also that ( A . H ) is detectable. Then the closed-loop system (5.3) sat isfies IITwlloo :S Q for any act uator failures within a prescribed set \l of susceptible actuators .
< 0 lI'i ll complete the p roof.
From (-1.14) and (4.1.5). ,ye obtain AVI = AVl and fi vI = fi ) is assumed detect ab le. this implies either R e(.\) < 0 or VI = O. If lJ I = O. t hen (~ . 1· 1 ) gi\'es
0: since ( A,
In the abo\'e formu lat ing the regulated output z reta ins all co ntrol sign als, incl udin g those associated with fai led ac tuators. For the case where such cont rol sig na ls are excluded from:; the fo lloll' ing resu lt holds:
(-1.16 )
243
Theorem 5.2. Lel X ~ 0 satis(l·
ATX + .\"..1.
+ -;XGGTX
- X B!i Blx
+ H1H
prese nt ed here ca n be eas ily extended to a combination of sensor and actuator outages. Certain interesting problems. howe,·er. cannot be directly treated within the dewloped methodology ei th er because the result s are too weak. or a different problem formulation lllu st be sought. Th e case when a st ate meaS Uf('IlH'nt is lost in sta ti c s tate feedback con trol fall s in the first category. since the problem effectil·eJy reduce, to a static output feedback control problem: while a s u ff\cicnt cond it ic)n hlr the ga i n Fna t rix can be sta ted. it takes th e form of an ('xtended .\ll E[.'i] for which cxistcnce of so lutioll" i, not g uarant epd for aliI· o. TllP sin gle outage problem i; all illstance vf the later: in this problem anyone. but only OI1t', ~(·11S0 r. for exa!llpk. call ('xpcrit'l1c(' all outage at sOlne tilll<' . ilnd tll(' goal i:3 to prol·i c!P guaralltC'ed performallce Ull til IIH'i\;urcs are ulldcrlak,'n ('ith"r to r"pla ce the sC'nsor or the controil"r is redesigned u:
= 0 (5.6)
()
A SSllme
(.-1.//) is deFeclable. Fhell Ihe closed-loop
\ .·1 .l. I); /,.; i.r
J.
(;:)(;1 -
+ (;11" .
- -
S-,"SleIlF
c:).,.
.-1 - IJ; /,.; i- I(;
1{,'ferCIICt.';
u .-1
.)
- I
:2
(J
·1 0
8 0
U -I I 0 .) - I 0 0
0 0
(; =
.)
I 0
fI
llj aSSUI11C'c\ s('ct!n~
tll
I l j~(s) 1 1
and
< 0.· 1.) C
·1 0
13 1
I
()
0 0 0 0
0 0
[ IJ .I . ..\ckerlll,,"n. S'lIlIfJlul-LJIl/a ('oll/ral SY8/U118. Springer. \·cr lag . H,'idel lwrK. 19 ,~ ej.
Pi
0 0
Y. J. (,h o. z. Hi ell. and D. 1\. l\il11. Roliable control \·ia "dditi,c redundant adapti,·c control. In Procadings of /he .·I I11,rimll ('oo/ml COllf,r,ore. pages 1 .~ 99 190~. Pitt sburgh. 1'.\. IV';').
[:l] H.. \. Ilat c and .I. Il. ("holl" . ..\ reliable coordinated de·
I
rClltr" lifl'd control s,·stcm design. In Prrxudillgs of IhE .!Slh CnllfuulCl 0 11 /)(I·i.,ioll Illld COll lrol. pages 129.')- 1300. "[""'lIpa. IT. i)PCt'1Il1wr 1~)'~9.
0
~1
0 0 0 0 0
with
0
0 0
2 - I
:2
0 U
10
[.1] J. Dm·lo. 1\. (;Im·cr. 1'. I\hargon eka r. ane! D. Francis. Statesparl' sol' ltiollS to standard Ho and H x control problems. IFI-:F Fmn.,"clion.< 00 .-1 11/01110Ii( COII/ro/. A("· 3-1(8):831-
II~ ullr('iiai)}f'.
gUilriUllc",·d
It call be shO\\"1I usillg cOlllr
~ · II.
/'·1
-[-1-1.0:3882.830 - 8 , 8691:38. :308 - 18,.·190] -[0 :3.328 - ~J:3.' ·I.j 10:3 .8(;, - 159 .,0 7221.:301]
1'·2
and results in the c1o,cd-loop {-751.9 1. -~. 7·1
~p("ct
In~().
[e:;J \\ .. E. ll opkill s ..J. \Icdanic . end \\ .. R. I'erkin s. Output fp~dhack pole-plarc lJl en t in the design of suboptinrallin car fJII"tlratic rq,;ulators. 111/. J. C,,"lrol. :).1(3):·)93- 6 12. 1981.
nIm
[6] S. \1. Joshi. Failure-"ccollllllodating cOlltrol of large flexible
± j2.68. - ·U6 ± j2.9}
specccraft. [n Procc((lillg.< of /he 1986 cl merican Control COllfll'(llcL pages 1% 161. Seattle. \\',-\.1986 .
if both control are operatiolla l. and in the spect rum
[I] \ 1. \ Iariton and P. BCrlrand. ImprO\·cd Illultiplex control {-:122. e) . -6.1 ~6. -:3:318. -2.76 ± jU~} if
ll2
dntam ic reliability an d stochastic opt imality. 111/CI'/la/iorwl '/ourllol of Con/rol. ~ ·.j:219 - 23~. 1986.
fails . as compared to the open-loop spec trum { - 6286. 1,-12
with gain
s~·stellls:
112
[8J .I. \ .. \ Ieda nic. \\". R. Perk in s. and R. J. Veillette. On the design uf rel iable rontrol s~·stems. In ProcF
± j3. '0 8. -1.599 ± j:J .2'i()}
now rcdes igned using (.'i. -I) and (5.10 ) resuits in the
F;,
[9] D. D. Siljak. Reliable rontrolusing multiple control systems. Illlem a/ionalJournal of Cnlllrol. 31(2):303- 329.1980.
= -[n.0:3 - 6.18:1 82.-'i2·5 - 11.).68 199.63]
[10] R . .J. \·pillett(' ..1. \ .. \ Iedenic. and \\". R. P er kin s. Robu st stabilization and disturbance rejection for uncertain systems 1)\· decentralized rontrol. In Procuriill'l8 of /hE W orkshop 0/1 COIl/rolof (·Ilrer/aill Sy.'/ems. Birkhauser Pub!. Co .. 1989. Series: Progre ss in Systems and Control Theory.
guaranteeing II T(s)ll-x < 0.3 \I·hen both controls are operationa l and a closed -loop spect rum
{-6-+.5. -38.6·5 - ,·56. -2648 ± j2.1-+~} .
[1 1] R. J. \·eillett e . J. V. \ Iedan ic. and \\ .. R. Perkins. Design of reliable control systems. In PrOCeEdings of th e 29th Annual Conference on Decision and Con trol. pages 1131- 1136. December 1990. Honolulu. Hawaii. Accepted for publication as a full paper in the IEEE Transac/ions on Automatic Control.
Conclusions The described reliable design trades off reduction of th e norm with reliabilit y of performance in the face of possible sensor or actuator outages. Acceptance of a subo ptim a l value of the Hoc performance in t he base case leads to an infinite family of cont rollers from which to choose to achie\·e other design goals, in this case re liabi li ty of a guaranteed perfor mance in variou s outage situations. The methodology
n""
[12J \ 1. Vidyasagar and :;. Viswanadham. Reliable stabi li zation using a multi-controller configuration. A utomat;ca. 21(.5) :.599-602. 1985.
244
DESIGN OF RELIABLE CONTROL SYSTEMS WITH GUARANTEED DISTURBANCE REJECTION PERFORMANCE J. V. l\1edaniC*, W. R. Perkins*t, and R. J. Veillette** *Coordinaled Science Laboratory. Unive rsity of Illin ois. 110 1 Wesl Sp ringfield Avenue. Urbana. 1L 61801. USA **Deparcmenl of Eleclrical Engineering. Uni versity of Akron. Akrvn. OH 44315-3904. USA
A bst ract. A met hodolog\' is presented for the design of control systems that remain stab le in the presence of a pre,pec ified subset of possible actuator and sensor failures . and that maintain a prescribed le\'el of performance (distlll'bance rejection) as measured by an H ",,-norm bound. Both central ized and decen trali zed control:, rtlctllJ'(~S can be treat ed. Thi s ne\\' methodology is based on a modification of the Ric cali approach 10 " ",- norm optimization. An cxam ple illustrates the procedure. E ey \\·ord s.
relia bility. robu st control. se nsors and actua tors. large scale systems where = is the controlled output. and .'I is the measured outpu 1. and cons ider the co nt rolle r
Introduction COl1\'entional f,,('dbilck conl rol d"signs for a mul ti-inp ut. multi-oulput plant may reslllt in IlI}Sal isfiIClory control system p('J'forman('('. or e\'l:n instahility. in the (,1'('111 of actuato r or se nso r outng('S. ;\ COIII rnl S~'SICIII d('sigllf'd to lolerale failures of actuators or of 5('n,ors \\'ithill a prcspecified subse t of all pos sible fai lures. \\'llik retilining de, irrci contro l s ~'slell1 properties. \\'ill be called a "r('li"bl,," conlrul sy,;tcm. This paper considers ll1('t hodologica I 1"'O(,l'rill n's for I hp design of reliable contro l sys tem s.
, (
u
z
= (H.T), U
Y= Cx+
Let the
where
Fe
A Ell ] [ LC A+G'lld+EI(-LC '
Then a key lemma on bounded real systems states that the resulting system is stable and satisfies
if ( Fe. He) is detectable and the algebraic Riccati inequality
has a solution Xe :::: O. Thus. an IJx-norm-bounding control will result if J(. I\d. and L are chosen so th at these conditions a re satisfied. The approach used for deriving reliable designs in [8,11] to express the Riccati inequality sufficient condition as
F; Xe + XeFe
+ ~XeC.C; X. + H; He + p.
= 0,
IS
(1.2 )
Q
where Pe :::: 0 is ullSpeci fied. To ach ieve rel iability, we pick p. such that the closed-loop system satisfies the key lemma not on ly when all control components are operating properly, but also in contingency situations when some sensors or actuators have failed. Cons ider a design that must tolerate particular sensor outages. Perturbations 6J.F. and 6J.H. are identified which correspond to fail ures of the susceptible sensors , so that the triple (Fe + 6J.F., C" H. + 6J.He) then defines the plant in a n outage situation. By adding and subt racting
(l.la) Wm
CO
.r, = F,.T, + Cew.
To introduce the approach consider the system
= kc + Bu + CWo
+ Bu + Ct;:o + L(y -
where t(·o is a model of the process di sturbance. resulting closecl-Ioop system be
There have been relati\'el:- fell' pre\'iolls attempts to develop methodologies for the design of reliable contro l system s. and these have had various reliability goals [12.2 ,1,6,7.9.3]. These designs hal'c not how('I'{'I' . addrl"ssed the isslle of pro\'icling guarantees on system pcrforma.nce. Thf' incorpora tion of performance guarantees \·ia a bound on the H"" norm, in addition to closed -loop stab ility. is a crucial contribution of the approach initiiltpd in [8.1 J]. Designs proposed in [11] arc characterized by closed-loop stabilit\· and H x norm bound when all control component.s are operating. as \\'ell as for thp case when any admissibl" control-component fail'lres occur. In the centralized case. admissible failures co",isted of any actuator or sensor outages within a predefined se t of susceptible actualors or sensors: in thl' decentral ized ca'e. admi ss ible failures co nsisted o f any controller outages within a predefined set of susceptible co ntrol channels. The C'xistence of appropriate solutions to the developed design equa l ions is sufficient to guarantee the rel iable stabi lity and performance of the closed-loop system . The reliable designs el'oh'ed from the basic H"" designs given in [4] for the cent ra li zed case and in [10] for the decentralized case. In this paper \\'1" will consider additional types of sensor/actuator outages. and the loss of actuators and of state \'ariable measurements in the case of state-feedback controllers.
.i:
.-\ ~ },' ~
(1.1b)
tThis work was supported by th e Joint Services Electronics Program under Grant N00014-90-J-1270 , and Sundstrand Corporation.
239
appropriate terms. t he co ndition (I.?) is th en r('written in th e form
( F + ~F,)Ty, + X ,( F, + ~F,) +-;!-r X ,(;,(;; X , + ( He + ~H, )T(H, + ~H, ) = -P, + ~ F,TX, + .\, ~F. + H ;r ~H , -L ~I/ T 1/. -'- ~H.T ~1I .
Remarks. 1. \\'ith al l sen sors operatio nal. which corresponds to w' = 0. T:Js) = T(s) is the nominal closed-loop transferfUllction matrix from n' to z. where
1\.:3 )
.-\ specific P, ~ 0 is then drtrrlllinNI ,u,h that tllf' right-hand , id,' of (I .:l ) is negatiH' "·In i-ddinik. T helt . 11 .:\ ) imp lie,;
(F
+ ~F,)T X , + X , I F. + ~r ;)
+:f,X, G, G; X,
+ ( 11, + ~II , )1(1/ , + ~H ,) :S
O.
Theorem 2.1 cOl'ers this ca se automatically. since ~. = 0<;;; n. If senSOrS corresponding to a nonempty s ubset ~. ;: 11 fail. then T~(s) is the res ultin g transfer-function matrix from tc w to z . where
1 l.! I
so that tile perturbed S.I·'tC'm ""ti,fil'" t11t~ prillcipa l hypoth C's is of the ke.l· Iemllla. Ik"iglt "'1 l1atio ll " d('I'eloped ill :~ .ll l us in g thi:::i app roac h fu r th e <:il~(' ut' ~(' II!"or olltag{ ':-\ alld fo r
U' -. =
....
t he case of act ua to [' :, (,)u tag(':-; arc ~lI l 11l 11 a rized in t h(' 1\(':\ t
U'o ) (
.
tc m .:·
with IC", -, co ntaining only those components of mea surement noi se assoc iated with operational sensors. Tl lm . fo r the re liability formulati on in [11]. a se nso r failure effect il'ely e lim inates the associa ted se nso r noise. Th e p roblem whe re th e sen so r noi se pers ists although th e plant meas uremen t has b een lost will be treate d in Section :l.
sect ion. The Design Equatio ns for t hf' l3 a, i, :;('n,or and .\ ctllato r Fa il url' \I od,", Sensor outages .)
n <;;;
{I. ? .... dim l lil} corre'IJ()ltri {() a ,e!cn"d ",Il'(,t of sc nsor, s usc(' ptihl, ' to ollta~es . Introd uce the d"l'O lliposition ( ' = ( \2 + ( 'u . 12.1 ) 1. ('1
1l1 <-tt rix
f-1+ =
l ",
formed from C by zf'roing out rOIl's corresponding to SllS(,(,Ptible se nsors. 1.(' 1 .... S; 11 d('not(' s('n"ors that aCIII"lIy eXI)('ri e nce an outag('. and let TJs) den ot(' the trall sfe r-fun cti o n matrix of tll(' r('s lIiting clos('d- Ioop 51·, t(,l11. It is COItI"'lIi('lIt to adopt th e notation
=0 2
(2.:3)
CC + L",C", .
(:Z . I )
C~0Cn
0) > 0 - 0 '
~o
12:1)
+ I. ~.
(
Pl'OoI In lif'II' of Remark (:3) abol·e. if (2 ..5) and (2.6) have appropria t I' sol "t ion s. [-11 (2.8)
sat isfies
lI'here C", and C_. han: Illeanings ana logo ll's to tho s(' of Cn a nd Cn in (2.1 ). Since w' S; n. C~ C'" :S C~ Cn . Also d('colllpose the obsen-er gain a s
L = L~
C~J.
Th is identifies the matrix P, in (2 .2 ) as
wh e re Ct ! dC!l o l,'S Illl' ItIl'ihUr<'!lll'1ll Illat rix i1s."ociatcd wit h 11. a nd CIl denotes the l11ea SUI'I'llll'nl matrix "",o,iated witll t he complemen lary su bs(' t of m('aSllr('nwnt s. Till". Co is
c=c+c.
Th l' design equations (2. cj) and (2.6) can be I'ie wed as ari,ing frolll th e repla cement of I-I by th e aug me nt ed
(2.9 ) with }' = (;!, Xl + 02X) . .\10reol·er. it can a lso be show n that ( F, . !-I<+) is a d etecta ble pair , wh e re
so that
LC =
(2.10)
(That is . L: ha s zero co lum ll s cOlTl'sp()lldin~ to ,"mors I"hiell hal'e actua ll y failed.) TI1<'n thl' followin g r(,su lt Il old s:
Thi s guarant('es stabil it y an d the !-I x· norm bound 0 for the ca se no se nso r out ages when the syst e m is described by the matrices ( F, . G, . He). For se nsor outages corresponding to w' <;;; 11. t he controller dynam ic structure is not affected by a se nso r outage: only the controller input structure is changed . and the controlle r bf'co m es
Theorem 2 .1. A ssume ( A. f-1 ) is df'tpcta/Jle. ilnd/et X ~ 0 and }' > 0 satis(1' th e .·l /gf'iJrai, Riccilti Equations (.-IRE,)
ATX
AY
+ XA
- XBBTX +;!,XGCX + HT l/
+ YA T + },YH T H\' 0"
+ 02ClCn = O.
- \ 'clCn )'
+ GG T = O.
(2.'»
(2.6 )
Gil'e n 12 .11). the closed· loop sys tem matrices become
respectil-'ely. and the side condition
_ BT.\.- . X d = -;!-r GTX. (J - o2Xn-l} ·CT .
(2.12 )
(2.1)
G,,, =
lI'ith A + (;1\d + Bh' Hurll-itz. Then. (or sensor outages corresponding to any w' <;;; 0.. the closed-loop s.l-stem is sta ble. and IIT;;;II"" :S o.
from which it follow s that
(2.13a)
240
be (2.1:3) in (2.9) and the fact
F/~ ..Y, -i- S , F~ " -'-
CAC n 2: C!C
(2.136)
A ssume t he controller is open -loop (in rem a lly ) stable. Then , for act u ~ t or outages corresponding to any w' <::: 0 . the closedloop system is stable. and Il f"", ,, ::; o.
(2.J:3c)
The proof is omitted due to s pace limitations. It proceeds parallel to the proof of the pre\iou s theorems: see [11) .
lead5 to
R em ark s.
7X,r; ,J;;CX" + f(I fI , =
-c,r..J:_Y, - SJ ,J',_
1. For actuator outages corresponding to ~. <::: fl . T~(s) is the transfer-function matrix from (e to c:;,. where z:;, in cludes only those control components associated wit h operational actuators. Thus. the control signa ls asso· ci~ted with failed actu~tors are dropped from the regulated output c. and do not enter into the computat ion of the H x norm. The formulatio ns where these s ignals are retained in :. a lthough tht' can t rol loop containing the actuators are open. is consider(·d in Section-l .
-?\J ,J;~ X, - ()2 (cl) (C n 0 ) ::; -C}~L;_S, - S,L . f ,_ -?Y.L u L;".\", - ()2C/_C_ = - (~X, L , " + CleL) u:e~.Y, + oc,_) ::; o.
Hence. if (F,~. H , ) is a detectahle pair. then F, " is ]-I\lJ'w itz. and i :. \ <, ) = II ,(, I - r ;:,)- I(;.:_. the trall sfcr·fllllc t ion Inatri~ fronl IC" to c. satisfies IIT:!I" ::; (). The dc-tel"lability proof is routinc: If (.1 = (e? l'j) i 0 sat islic,; I ;J = ,\t· and flet· = O. then Ael = .\t·) alld 11 Cl = O. with (.-\. 11 ) assumed Cl detectable pair. Therefore . ('ilher F! c(.\) < 0 or Cl = O. If VI = O. then }< ~e = I ~~e = ,\e and 11 , ," = 0 gi\·cs H,+ ," = O. Since (F~ . He+) is a detectable pair. /i t(.\) < 0 Q.E.D.
2. T he design equations (2 .16) and (2.17) call be \·iewed as arising from the replacenwnl of C \\·ith the augmented rnatri~ G+ gin'n by (:2.1~). Thi s corresponds to choos-
.lflg P, =.\" , ( Bo0B~
0) 0 .\" , 2: 0 .
Actuator outages Sensor Outage '.l ode with Res idua l '\ lcasurement \"oise
Let 0 <::: {l. 2 ..... dim(u)} co rrespond to a selected su bse t of actuators susceptible· to outages. Introduce the decompo-
Consider again the sensor outage problem. but assume that the output chanllPls are always acti\·e and when a meas ureme nt is los t the sensor still injecls measurement noise into the sys tem. Thus. in a failure lllode
sition
B = Ho
+ l3 0 ·
(1. J.J )
where Bn denotes the control mat rix a ssoc iat ed with the set 0. and Bn denotes th e control nIatri~ ao;sociat('ci with the complemcntary sllbsct of control inputs. Tlltts. R {j is formed from B . by ze roing out columns corrcsponding to su,;,"cptible actuators. Let ~. <::: n correspond to actllators t hal a ctually fail. and let T:,.($) denot(' Ill(' transfer-fllnction rllat ri~ of the resu lting closed-loop sys tem. It is con\·cnient 10 ~dopt the notation (2.15 ) B = 13.., + 8.., .
y, y,
Ci,l"! IC,.
+
ll'i-
i
E~·
i ~ . .;.,' <::: rl.
(3.1 )
\\·het·" the mea surerne nt noi se yector is decomposed as U'~ = [U·I (CZ .·. IC, ) . The controller in a failure mode is t.hen
where B~ and B ~ ha\"(' meanings analogou s to those of 130 and Ro in ( 2.1~). S ince~· <::: n. RJJ! ::; /J,, 13~·. c\ lso . decompose the feedback gain as
U
}\.(
~
(A + BX - LC + GJ\"d)( + !. ",y +
L~w m
(3 .2)
so t hilt t he closed loop system is characterized by
so that
. (GO) 0 L .
The o rem 2. 2. Let .\. 2: 0 and l - > 0 sat isf\ the AREs
AY
••T
+CC
+ 0 Z E oBoT
=
(HO) 0 I{
.
(2.17)
o.
respecti,·el.\ ·. and the side condition lT~axfXI ·) < define
He =
:\ote tha t retaining the measurement noise associated with failing sensors means that G, remains as in the base case. Transforming to the {1·1, d coordi nates produces now
+ y,.\T + J,YHTH Y - l -CTCT o
G, =
Cl'.
Also
(2.18b) and let the controller be
(2. 19a) u=
J{~.
(3.4)
(2.19b)
241
We again seek
Pe
in addition /\ d is defined as in (3. t:l ). Fin a lly (3 .10c) reduces to (3.12) when L is defined by (:3.1 -1) \I-ith 11 = ,,'XI-I.
so that
Th(" required detect ability condition on ( l-~ ~ , H ,) is established exact ly as in the case of Theorem :!.l.
Y,
p05Sesses a solut ion 2: O. Assuming a block diagonal solu tion as in (':' .8) and using (:1.-1) allol\"s (:3.,3) to be rearran~ed as
r/~ Y,
\01\"
+ Y, r~ ", -r- -f,Y, C~',(;; _\7, -i-1T[H, = -Pe - CfjLY, - _\:JuC~
It is not ed that the ."REs (:3.11). 1:3.1':' ) that ,wed to be so h'ed in sequenc(" can be replaced b,' two completely decoupled algebraic Riccati equiltiom'. Spe<·ifically. the _-\ HE (:3.1:2) can be replaced by _-t} -
+ }- ·11
-
if
I qC[) }- + f,lll'/f} + } CAC n }- ~ (;(;1 = 0
(31;j)
\I'ith \\- and}- related by 1-' = \1 '-',,,'.\: Ill<' reqllirelllent 11- > 0 in TI"'(""'111 :LI del11and, tililt tl\(' sid" condition ~2 > IT,~""I Xl ) lw illcllldf'd . .-\n .\ It<'rn,ll iw' .\('\IIi1tor-Olltil ,'!.I' \l"de
lL\;' + \ , /~c + f,-\ ,{;,{,-;5:, + Ji? 1-1e
s - (nC}_.j. +X,f_,~)(etC," + +r;~X,):
so. by prnioll> Mgllme'IIb. I he system I\"ill be SI able and haw' gllaran!(:'I'd 11,- -nurn, I> OllIH.l et for th e considered failure mode prO\'i c!cd (:l ..'») is sa ti sfied.
it
We now u se the slIfricicnt conditions (:3 ..5) and (:3. / ) to determine thp corr('sponding design equation s . First note that choosing as
P,
\"'hen all aclllator fails, we may still "'ish to de!!land that Ilw control e ,wrgy of the faikd ilClualor I", bounded e\'t'n though t he loop is brokell. To this effect, W(' cOIl., ider the output signal to be li; = 0_ i E ~. ~ as far a, th!;' system dyn.aJ11ics are cOllcflrn('d. but aSSllIlW that Ili = ]\·lE.. i E ..t-' as far a, the regulilted outputs are cuncellH'd.
n_
Fora giH'1l failure mode_ the cOlltrolle rtll('1I ha s th('structure (1.1 )
(:39) gllaranleps tl,al (:\./) ",ill hold for "ny f"ilurf's~' ~ O. ),h"lI. recalling tllf' block-diagonal st rncture of Y,. and parlit.ioning (:l ..~). \IT oblain Ih" Ihrf'c Ctl11dilions X(A
+ RJ,') + (A + + Il T/-{
Ell\ )' X
+
,+,XC;C;TX
+ 1\11,- + 02'C·0.Cn
= 0
with !\. !'-d+ ilnd L to he dc\r'\'IlIilwd. TIlf' clos"d-Ioop systelll is tllf'n characterized b\· Ill<' t ripkl
(-1.:2 ) (TI0o)
. ((;0) 0 [
('e =
+
X,(.-I - LC G /'-d) +bX,CCTX,
+ (A -
LC
+ GI'-d)TX,
+ -;}rX,UTX 1 + I,-IJ,+bX1InL'{;_\-1
.Ii, =
(110) 0 /,-
or after a coordinate transformation by the triplet
(:3 .101')
= 0_
A. suificipnt condit ion for this to hold is captured in the fol-
( 1.:1 )
lowing Theorem: Theorem 3,1. S'lppose (A. H ) is detcctabJf'. and Jet X 2: 0 sat is(\' the ARE ATX + XA - XBBTX +c!>- X GC;TX + HI H + o2C6'Cn = O.
and ll" (~1
(3.11 )
\011" assul11ing a (:3 ..'i) -type condition holds for the case of no failure. it then folloll"s that in a failure Illor\t'
R",( _\(, )
> 0 satisZ\' the .-iRE
+ GJ\d)W + IF (A + Gl\d)T -
II"C0.Cn l\'
+bli-1,-T IdF + GG T = o.
:=::
I~1;_\(, + Y, F~" +7X/;-,C';X, + 77;77, -P, - X,BuK,~ - K,r.i3L-\;',
(-1-1)
(3.12) Therefore. choosing
P,
to guarantee
\\-ith (:3.13) Then. for senso!' outages corresponding to any ",' ~ n. the closed Joop system, ,,-ith the controJJer and observer gains
is stable, and
I{ = _BTX. IIT",II."" < 0_
L
= WCT
\I-ill result in a reliable design. It \I'ill noli' be demonstrated that choosing
(3.14)
Proof. Equation (3.1 0a ) reduces to (3.12) when f{ is defined as in (3.14) . Condition (:3.10b) is identically satisfied when
242
leads to a computationally fea sible reliable design. This follows since , for P, as in (-1.6). equation (-1.-1) becomes
~ 0
lI'hich is a sufficient condition for 11 T.oII ", to hold.
a nd 11 [ 11 x
lI'ith '/2 # 0: t hat is. ,\ is an eigen \'alue of the ope n-loop controller dynamics . Thus . if assumption (i) holds. then Re(A) < O. Suppose (i) doE'S not hold.
~ 0
The design cCjuiltions can be dcr i" ed from (:3.,,)) and (~.6) . .-\SSUIllC agii in I to ha"c the block-diil!:>;ona l >trllct ll rc (:2.S). TheT! from (·1. 6 )
j'J,
SincE' (-I I:")) gi\es
I\-'/2
= O.
(~. IG )
gi" es (4.1 j' )
:'II ultiply ing (-I.9c) on the left by v;. and on the right by V2' ilnd llsing (-1.17) and the re lalion XIL = 02C 1'. lI'eobtain
=
(-I.IS )
If inequalit\· holds in (-I.IS). t hen Rtl .\) < 0: if ('f]uil lit\, holds. then C"2 = O. ilnd XI > 0 implies Rd.\) = O. To 1'1'('rill(k tl'is 1'\I,sil.,ilily. eith ... 1' as'tl111ption (i il 01' (i ii ) is surtici(,11!. sincc' (·1 1j' ) lI'ith ('''2 = 0 .Hld 1""2 = 0 gin's
and (:3.-5) can 1)(' partitioned into the threE' condilions .\ 1.-1 -;- J3J, ).l. ( .-1 + lJ J, )' .\ -;- /IT 11 -+- [,'1' [,' ..;.. .\' R" 13~.\
X fJ [,' -i.\' I
( ,-I
+ -;'.\(;C;T\ T
'.! [,',~ [,' f' =
n
J,'3~ C:T ' \ 1 - -"-: .W;(; F \ I - .\ :f3,:r-J?1.\'1 :.;. [('[,' = 0
+ ( ; + [," !-i-
- (. L ) -+- ( . 1 + (;~ [,'f L )J .\ I +7.\'1(;(;T\1 + -f;'\ILi.F\1 -+- J, ' [,' + XIH,:/J~X I + '1.1'·fi ['-n = n.
where (;+ is gi"eT! in
T
-
1 1.\),,1
(.1. ~ 1(, )
(.
(~ .9(·)
Consider no\\' t llf' statf'-rt'e dback con t rol problem of finding co ntrol lall' 11 = !\'.l' lI'ith tl1<' obj('cti\'(> to guarantee a prescribed fi x-no rm bound for th(' transfn-fu nction matrix from /I' to : = (~,"). d('spite any actuator outages lI'ithin a S('il'ct( 'd sul,s"t (l illl a\'ailabil' actuators, .-\ ss u1l1ing a state kec!back con t ro l. t he closed-loop syst<'n l becollles
(:2.I~).
it
T h eo r e m 4 .1. .. \ SS IIIll(, ( ·1.11) i, cietertable' ilnd Ie't .\ :::: 0 sat isf,' lite ..l, RF. . IT.\'
+ X.-I -.\' 1J(!Bi!'\
+:2 .\' IJnB~X
+~XC(;I X
+ N TH
= ()
i l.IO )
.r
I(s )
+
+~ H' 1,'f~ [,'0 I I ' C;(; 1 0' l3nlJ~ = U
+
(~,II
(A
)
(-1 .1'1. )
is st ablc. anci II T.A S)II x <
Cl
fnr \I'it h am'
(.·1 + J3 1, )Ty
-
. 1 + J3", 1\':;1x + (ffT 1,-J)
<:;; fl fai l. th" closed-loop
:
= (::).1'
A - B:..J'·.X 1 (;
+ X(. ·\ + B X ) + ~ X(; (;T X
m+
Pro o f. It is pasily sho\\'n that choosing X as in (,1.12 ) rpduces (-1.9a) to (-l.1 0) whence choosing 1{'1 + as in ( 1.21) identical ly sa ti sfies (-l.9b ). Finally choosing L as in (-1.1:1) \I'ith II' = 02X I- 1 . rf'Cluces coTiClition (·1.9c) to (·1.11 ) . It remains onll' to sh oll' thilt tll(' 5\'stem is detectable. To tlli s pnd. Irt I1 ~ (~:) # 0 sa t isfy ,
X (.-I
+ l3j,' N) + ~Xc;c;1'.y
= - p -
[,.! B~ x -
(.5. :3)
X B~ 1\·~.
(5 . ~
)
[,.!
If P + B~ x + X R~ X~. :::: O. then (.5.-1) is a sufficient condition for the system (.5.~) to satisf\' the H x -norm bound. Ch oosi ng P = I\'l I,'n + X Bn BTz X is sa t isfactory. and choosing /\' = -BTX . lead s to the design equation .-ITX
+ X A + ~ )( G(;T X
+ X B n Blx - XBnB'hX
Sholl'ing t ha t Rd A)
(:3. 1 )
whne (.-\ + 1]],'. (~)) mu st bp detectab le. and f' :::: O. From C'd). II'P obtain
<:;; fl
( ~.I -I
= (;:').1'
C;
+(11T 1, T)(;n + p = O.
( 1.1:3 ) ~.
se t~·
I
T Ilf' sufficient condition for (.'d) to b(> lI x -norm-bounding for Ih(' nominal ,ystCIll is t hat some X :::: 0 sat isfy
Then . if (i) I he COIII roller is st aiJlc or (ii ) .. I + (;+ /\'.1+ + Jj 11' i., I-I u}'\"itz 0' (iii) .-1 + (;+/"J+ has no j~··i1.'\i-, cigcll\,i1ll1c,'. I}",· closed- loop s.,'slem \I'iUI controller gain (4. 20) and o/)serl'(:" gain L = II C T
:
+ Uj,'",):r + (;(1',
(;~)(s [
\\'ith 1\- = -B'.\-.
+ J-J!\').1 + ( ; U'.
(:n (,[ - .\ - B [\- )-
If actuiltors assoc iatf'd with a syst(, lll bCCOllWS
(A + G+ 1,'d+) 11' + I I' ( A + (;+ 11',1+) + +,- 11'[,' '/ 11' 11'
_ II'('T(' II'
( I
+ fl T fi
= O.
(.5.5 )
)
T he detectability is easily \·e rified. anc! so the design guarantees the desi red reliable disturbance attenuation. The following thE'orem summarizes the result.
(1.1.5 )
Th e ore m 5 .1. Let X :::: 0 satis f\ ' (5.5). and let 1\ = _ B T); , .--I ssume also that ( A . H ) is detectable. Then the closed-loop system (5.3) sat isfies IITwlloo :S Q for any act uator failures within a prescribed set \l of susceptible actuators .
< 0 lI'i ll complete the p roof.
From (-1.14) and (4.1.5). ,ye obtain AVI = AVl and fi vI = fi ) is assumed detect ab le. this implies either R e(.\) < 0 or VI = O. If lJ I = O. t hen (~ . 1· 1 ) gi\'es
0: since ( A,
In the abo\'e formu lat ing the regulated output z reta ins all co ntrol sign als, incl udin g those associated with fai led ac tuators. For the case where such cont rol sig na ls are excluded from:; the fo lloll' ing resu lt holds:
(-1.16 )
243
Theorem 5.2. Lel X ~ 0 satis(l·
ATX + .\"..1.
+ -;XGGTX
- X B!i Blx
+ H1H
prese nt ed here ca n be eas ily extended to a combination of sensor and actuator outages. Certain interesting problems. howe,·er. cannot be directly treated within the dewloped methodology ei th er because the result s are too weak. or a different problem formulation lllu st be sought. Th e case when a st ate meaS Uf('IlH'nt is lost in sta ti c s tate feedback con trol fall s in the first category. since the problem effectil·eJy reduce, to a static output feedback control problem: while a s u ff\cicnt cond it ic)n hlr the ga i n Fna t rix can be sta ted. it takes th e form of an ('xtended .\ll E[.'i] for which cxistcnce of so lutioll" i, not g uarant epd for aliI· o. TllP sin gle outage problem i; all illstance vf the later: in this problem anyone. but only OI1t', ~(·11S0 r. for exa!llpk. call ('xpcrit'l1c(' all outage at sOlne tilll<' . ilnd tll(' goal i:3 to prol·i c!P guaralltC'ed performallce Ull til IIH'i\;urcs are ulldcrlak,'n ('ith"r to r"pla ce the sC'nsor or the controil"r is redesigned u:
= 0 (5.6)
()
A SSllme
(.-1.//) is deFeclable. Fhell Ihe closed-loop
\ .·1 .l. I); /,.; i.r
J.
(;:)(;1 -
+ (;11" .
- -
S-,"SleIlF
c:).,.
.-1 - IJ; /,.; i- I(;
1{,'ferCIICt.';
u .-1
.)
- I
:2
(J
·1 0
8 0
U -I I 0 .) - I 0 0
0 0
(; =
.)
I 0
fI
llj aSSUI11C'c\ s('ct!n~
tll
I l j~(s) 1 1
and
< 0.· 1.) C
·1 0
13 1
I
()
0 0 0 0
0 0
[ IJ .I . ..\ckerlll,,"n. S'lIlIfJlul-LJIl/a ('oll/ral SY8/U118. Springer. \·cr lag . H,'idel lwrK. 19 ,~ ej.
Pi
0 0
Y. J. (,h o. z. Hi ell. and D. 1\. l\il11. Roliable control \·ia "dditi,c redundant adapti,·c control. In Procadings of /he .·I I11,rimll ('oo/ml COllf,r,ore. pages 1 .~ 99 190~. Pitt sburgh. 1'.\. IV';').
[:l] H.. \. Ilat c and .I. Il. ("holl" . ..\ reliable coordinated de·
I
rClltr" lifl'd control s,·stcm design. In Prrxudillgs of IhE .!Slh CnllfuulCl 0 11 /)(I·i.,ioll Illld COll lrol. pages 129.')- 1300. "[""'lIpa. IT. i)PCt'1Il1wr 1~)'~9.
0
~1
0 0 0 0 0
with
0
0 0
2 - I
:2
0 U
10
[.1] J. Dm·lo. 1\. (;Im·cr. 1'. I\hargon eka r. ane! D. Francis. Statesparl' sol' ltiollS to standard Ho and H x control problems. IFI-:F Fmn.,"clion.< 00 .-1 11/01110Ii( COII/ro/. A("· 3-1(8):831-
II~ ullr('iiai)}f'.
gUilriUllc",·d
It call be shO\\"1I usillg cOlllr
~ · II.
/'·1
-[-1-1.0:3882.830 - 8 , 8691:38. :308 - 18,.·190] -[0 :3.328 - ~J:3.' ·I.j 10:3 .8(;, - 159 .,0 7221.:301]
1'·2
and results in the c1o,cd-loop {-751.9 1. -~. 7·1
~p("ct
In~().
[e:;J \\ .. E. ll opkill s ..J. \Icdanic . end \\ .. R. I'erkin s. Output fp~dhack pole-plarc lJl en t in the design of suboptinrallin car fJII"tlratic rq,;ulators. 111/. J. C,,"lrol. :).1(3):·)93- 6 12. 1981.
nIm
[6] S. \1. Joshi. Failure-"ccollllllodating cOlltrol of large flexible
± j2.68. - ·U6 ± j2.9}
specccraft. [n Procc((lillg.< of /he 1986 cl merican Control COllfll'(llcL pages 1% 161. Seattle. \\',-\.1986 .
if both control are operatiolla l. and in the spect rum
[I] \ 1. \ Iariton and P. BCrlrand. ImprO\·cd Illultiplex control {-:122. e) . -6.1 ~6. -:3:318. -2.76 ± jU~} if
ll2
dntam ic reliability an d stochastic opt imality. 111/CI'/la/iorwl '/ourllol of Con/rol. ~ ·.j:219 - 23~. 1986.
fails . as compared to the open-loop spec trum { - 6286. 1,-12
with gain
s~·stellls:
112
[8J .I. \ .. \ Ieda nic. \\". R. Perk in s. and R. J. Veillette. On the design uf rel iable rontrol s~·stems. In ProcF
± j3. '0 8. -1.599 ± j:J .2'i()}
now rcdes igned using (.'i. -I) and (5.10 ) resuits in the
F;,
[9] D. D. Siljak. Reliable rontrolusing multiple control systems. Illlem a/ionalJournal of Cnlllrol. 31(2):303- 329.1980.
= -[n.0:3 - 6.18:1 82.-'i2·5 - 11.).68 199.63]
[10] R . .J. \·pillett(' ..1. \ .. \ Iedenic. and \\". R. P er kin s. Robu st stabilization and disturbance rejection for uncertain systems 1)\· decentralized rontrol. In Procuriill'l8 of /hE W orkshop 0/1 COIl/rolof (·Ilrer/aill Sy.'/ems. Birkhauser Pub!. Co .. 1989. Series: Progre ss in Systems and Control Theory.
guaranteeing II T(s)ll-x < 0.3 \I·hen both controls are operationa l and a closed -loop spect rum
{-6-+.5. -38.6·5 - ,·56. -2648 ± j2.1-+~} .
[1 1] R. J. \·eillett e . J. V. \ Iedan ic. and \\ .. R. Perkins. Design of reliable control systems. In PrOCeEdings of th e 29th Annual Conference on Decision and Con trol. pages 1131- 1136. December 1990. Honolulu. Hawaii. Accepted for publication as a full paper in the IEEE Transac/ions on Automatic Control.
Conclusions The described reliable design trades off reduction of th e norm with reliabilit y of performance in the face of possible sensor or actuator outages. Acceptance of a subo ptim a l value of the Hoc performance in t he base case leads to an infinite family of cont rollers from which to choose to achie\·e other design goals, in this case re liabi li ty of a guaranteed perfor mance in variou s outage situations. The methodology
n""
[12J \ 1. Vidyasagar and :;. Viswanadham. Reliable stabi li zation using a multi-controller configuration. A utomat;ca. 21(.5) :.599-602. 1985.
244