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Robust Servo Problem for Decentralized Control Systems with Partial Parameter Perturbations
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ROBUST SERVO PROBLEM FOR DECENTRALIZED CONTROL SYSTEMS WITH PARTIAL PARAMETER PERTURBATIONS S. H ara
Ahstract. This paper is concerned with the partially robust servo problem for decentrali:ed contro l svs tems with partia l parameter perturbations in the case where the outputs of each con trol agent are required to track different types of reference commands ~ithout steady-s tate errors. Necessary and / or sufficient conditions for the sOlvabilitv are derived. These lead to the fact that conventional solvabi l ity condition for th e robust decentralized servo problem presented by Davison et. al. is only a sufficient condition of this problem. A synthesis procedure of decentralized servo svs t em. ~ith matrix opera tions is proposed based on these solvabilit y conditions. It is also shown that the total order of the designed servo compensators may be sma l ler than that of the conve ntiona l one. Kev~ords . Se rvomecha nisms; Decentralized contro l systems; Large-scale systems; System anulvsis; Contro l sys tem synthesis; Parameter perturbation
I\TROnUCTIO,<
to the fact t hat the conventional solvability condition for t he robust decentra l ized servo problem investigated by Davison et.al. (1976, 1979, 1982) is only a sufficient condition of this problem. In Chapter 4, a synthesis procedure of partially robust decentralized servo sys tems is proposed based on these solvability conditio ns. I t is a l so shown that the to t al order of the designed servo compensators may be smal l er t han that of the conventional one.
Recently, much attention has heen paid to decentralized control systems with the enlargeness of plants to be controlled and the advance of computers. Both analysis and s"nthesis for decentra liced control systems have been investigated by many researchers. In connection with the servo problem Ilavison et. a!. (1976, 1979, 1982) have proposed some useful synthesis algorithms for rohust decentralized servo controllers under the follo~ing assumptions: 1) All the parameters of the plant vary independ e ntly; 2) Al l the externa l signals such as refe r ence commands and disturbance inputs are generated bv sys tems with the same modes . These results are effective for decentralized systems with parameter perturbations, but they have sho rtcomings in that s tr ong restrictions are imposed on the plant for the robust servo problem hei ng solvable and that the order s of the compensa tor s designed are necessarily large due to above unrealistic assumptions. In practice, however, ~' e oftE'n encounter the following cases: 1) Particular invariant characteristics such as integra l characteristics in d.c . motors exist in the plant. This means that the plant parameters are not all perturhed, i.e. , some para~eters are fixed at zero, one, etc. ; 2) All the external signals do not have the same modes . For example in crane svstems, the position of the cart has to track the reference input of s tep type without steady -state error but the angle of the rope is onlv re quired to be regulated at zero .
Notation: alA), p[A ], and A+ denote the set of the eigen values, t he r a nk , and the seude i nverse of a constant ma t rix A, respec t ively. For a ma t rix A(e) which is a f unction of ~erturbed parame t ers e, A* and Pg [A(e) ] denote A(e ) and t he ge neric rank, which is equa l to the maximam rank, of A(e), respectively. C+ and C- represent the closed right half complex plane and the open left half complex plane, respectivel y. PROBLE~l
STATE~lENT
I n this paper, we consider a decent r alized control system wi t h partial parameter perturbat i ons [peAce), {B ( e)}, {C })
i
i
x = A( 6) x + Bl (e)u
l
+ B2 (8) u 2 (i = 1,2)
Yi= Cix
This paper , therefore , concerns Idth the ro bust sen'O problem for decentral i zed control systems with pa rtial parameter perturbations in the case ~here the ou tput s of each control agent are required t o track different tvpes of reference commands without steady-state errors. In order to avoid the complexit v of the description this paper only deals with the case where 1) the external signals contain no disturbance inputs, :) thE' direct pa r ts of the inputs to the outputs do not exist, 3) there are only t~o control agents . Results obtained here can be easilv extendE'd t o the general case. (See Appendix.)
(I. a)
(1, b)
where xERn is a state vector, UiERmi and YiERPi (i= 1,2) are an input vector and an output vector of i - th control agent respectively, and eER r is an unkno~n and unmeasurab l e parameter vector with nominal value e *. The reference inputs Yri (i= I, 2) are given by the outputs of a linear f r ee sys t em [ rtF, {Hi }) Fz
(a(F) c C+)
(2.a)
(i= 1,2)
(2. b)
~here ZERnZ, "r.ERPi (i=I,2). . 1
The error vectors, I;hich are equal to the measura ble vec t ors, of each control agent ar e defined by
The organization of this pape r is as follows: In Chapte r 2, a partially robust decentralized servo problem (denoted by PRDSP) is defined . Necessary and/or sufficient conditions for the solvability are derived in Ch3ptcr 3. These condi t ions lead
(i = 1,2)
He r e, i t is assumed that a pair (A*, [B ,B ]) is l 2 1195
(3)
s.
1196
rCI],A)*
controllable and that pairs ( ic l
z (0), '9' -xI (0), "1 -x (O), an d "I q. [ ~.. f:*) . 2 Especially, PRDSP is so l vable by static compe nsators, if there exist a positive number [ and Zi ) ( V x(O,
and
2
both observable. [Remark 1) Lr(F,{Hi)) is a general form of reference commands, si nce polynomial and sinusoidal tYpe reference commands which are often appeared in the practical servo synt hesis can be expressed by this sys tem . For example, I ) Step: F=O H.= I 1
2) Ramp:
F= [ : :]
3) Sinusoidal: F= [ 0
W
Ha r a I(
wit ~i(the order of f i)=o ; i=I,2 such that the above two conditions hold. IS requires the stability of the designed closed loop system in the case where all the reference commands are equa l to zero. OR means that al l th e outputs track each reference input I,ithout steadystate errors . La stlv, the set of fixed modes, which is an important concept for investigating the stabi lity of decen trali zed systems, is stated .
1
-w 0 [De finiti on 2 ) (Wang and Davison, 19 73)
We now consi der a decentralized servo system with dynamic compensators having constant coefficient l: i
(Ai ,Bi ,c\ ,D i ) A.x.
+
Ri Yei
(4. a)
u.
C.x.
+
DiYei
(4 . b)
1
1
1
1
where ~ . E Rni
p -
The composite sys t em made up of Lp and Li (i=I,2) is des cribed by =
~(~)~
(S . a) (i =1,2) ,
Y i = Ci x
and
(S.b)
_
1
=[
-B (8)C
2
-BIC I
Al
0
-B 2C2
0
/\2
C.
0
0
1
= {
s ! 11 9 - e* 11 < [ }
) . (6. b)
(7)
i:ed servo pr oblem (PRDSP) is solvab le at e=e* , if there exis t a positive number [ and decentrali:ed dynamic compensators ~i(~i,Bi,Ci,6i) ; i=I,2 such
v
ei (t)
=
0
; i==i,2
( ~8E
0.;)
(12)
(The outline of the proof) Conditions (I) and (II) correspond to the TS and OR conditions, respectively.
0
Condition ( I ) is a necessary and sufficient condition for stabilit,· by gain output feedback in each agent. Condition (11) i s that of a ce ntral i:ed sy s tem having an internal model of reference comTherefor e , PRDSP b,· s t a tic mands. (Hara , 1982) compensators is not different from the case of the ce ntrali: ed servo problem by gain output feedback . [Rema rk 2 ) that the property of internal stability is well-posed for parameter perturbation, s o that if the condition (8) holds only at the no minal value ~ = 9 * then the intern al stability condition i s al s o satisfied. Condition (T ) in Theorem I thu s consists of a condition a t the nominal value ~ * . Thi s i s true in the fOllOldng theorem s , lemmas , and corollarie s . ~ote
(8)
(9)
lik e th os e inv e stigated h,· Davison et. a1. ( 1976, 19 79, 1982). HOI,ever, the po s ition s are reversed, i.e., Lci are follol,ed b,· =si in Fig.1 but :: si are
(11) Output Regulation (OR)
ZiT'"!
['COlHI F
the so l vabi lit y condition s by dynamic compensators is discu s sed. Without lo s s of generality, a decentralized servo system in Fig.1 is considered he r eaft er. Th e control l er con s i s ts of s ervo compensators =ci for OR and sta b ili:ers =si for IS
that the follol,ing tl,O conditions hold (I) Int ernal Stability (IS)
t~oo
(11)
H2 .
[De fini t i on 1 ) When a reference command is generated by a system ! r(F, {Hi }) for a decentralized control system : p (A( e), {Bi( e ) i , {Ci 1 ), the partially robust decentral-
(y S€:::
2 * L B. K. C. ) c:. Ci= I I I I
'(O l ~~ T( 8)
, (6. a)
A partially robust servo problem for decentralized con trol systems is thus defined as follOl,s :
O(A( e )) C C-
+
l1
2
This composite system is required to retain two servo properties, namely internal stability (IS) and output regulation(OR) , for all parameters 8 contained in th e [ -neighbourhood at 8 = e * r epresented by
,,* .. [
*
(I I) There exist a posi tive number [ and an nxn z matrix T( 6) such that
_
A(8)- L B. (8)D.C. , -B (8)C I I i=1 1 1 1
C.
(10)
[Theor em 1 ) PRDSP is solvable by static compensators, if and only if the following two conditions hold (I) There exist mi xPi matrices Ki (i= I, 2) such that
T
2
A( 8)=
. ., * f B.I\ . C.) i=lll1
First, a necessary and sufficie nt solvability condition for PRDSP hy static compensators is proposed.
(A x
+
SOLVAR ILITY CONDITIONS rOR PRDSP
where x::;
*
{S;},{C ; ) .
No te that the in puts and
the outputs of these systems are measurable outputs and control inputs of each agent in L , r espective l y.
~
(A
i
( i =I,2).
1
n
{K. }
is said to be the set of fixed modes for Lp(A * ,
~
1
1
1
x. 1
* !' ( {C · } ,A , {B*I· }) =
~ext,
folloh·ed b,· ::ci in Da\·lson ' s controll e rs.
The
Robust Servo Problem for Decentralized Control Systems
11 97
contains the para meters of Lci to be designed. Thus, th e necessary and / or sufficient conditions represented only by the parameters of the given plant [ are proposed . p [The orem 2)
If PROSP is solvable, then the two conditions hold: (I) fI({C }, / , {S:}) i
Fig . l
If the servo compensators Lci (Aci,Bci,Cci,Oci) are represented by
x. Cl
A .x.
+
B
.u .
(13. a)
u.
C .x .
+
0 .u .
(13.b)
Cl Cl
Cl Cl
1
Cl Cl Cl Cl
where x . £Rnci and u .£Rmci Cl
Cl
then the composite
e-
(1 7)
(11) There exist a positive number an nxn z matrix SO(9) and mixn
Block diagram of decentralized servo sys tem
configuration of the servo system in Fig . l is same as that of the centralized servo system treated by Bengtsson(19 77) and Hara(1982). It can be expected that the weaker solvability conditions and the lower orders of the compensators are obtained by this method, since the invariant characteristics such as integral characteristics contained in the given plant may be available for the internal model in this configuration .
c
follo~ing
z
and matri-
ces Si (9) (i=I,2) such that A( e) Bl (e) B2(e) Cl 0 0
1
[SO( 9) -j _ [SO( 9) FjSI (9) HI
[ C 0 0 S2( 9) H2 2 (The outline of the proof) IS condition requires condition (I). The fact that condition (Il) is equivalent to condition ( 11 ) in Lemma 1 which must be satisfied for OR. 0 When condition (11) in Theorem 2 is satisfied, the solutions Si (e) (i=1,2) can be written by hi
d \
(9)
L f . . (9)Q ..
j=l
IJ
(19)
1)
and define qi as
system r(L ,{L . }) is written by p
Cl
(20)
(l4.a) (l4 . b)
where f (9) are independent scalar function s of 9 , ij and (21)
[Q·l'Q·2' .... · 'Q·1 h) 1 1 i
where x
=
(9)1) (i=I,2). (22) ihi We now get two equivalent sufficient conditions for PROSP by means of the constant matrices ~ . [fil (e)I,· ··· .,f
T
x
and
[Theorem J)
i) If conditions (I) and (11) in Theorem 2 and (Ill) There exist mixqi matric es Qi (i =1,2) such
,~[:: ;h;l}R'r:;:: 'e!~:grj :": ; :~}
(23)
do not contain the modes A of F A necessary and sufficient condition for PROSP is represented by the composite system L in the same manner of Theorem 1. (The proof is omitted.)
[Lemma 1) PROSP is solvable, if and only if there exist Lci (i=I,2) such that the following two condi-
or the equivalent ii) If condition (I) in Theorem
,
p[~)
tions hold : (I)
fI({e),A, {(})
c
e-
(IS)
(Il) There exist a positive number £ and an (n+n
+nc2)Xnz matrix fe e) such that [ A(e)
~~
1
T( 6)
~
[':;)F 1
C V 6 Ell;)
cl
(16)
0 This lemma, however, may not be useful for the design of Lci' because the condition of this lemma
holds and
(Il ) Condition (Il) in Theorem 2 holds with
qi
~
Pi
(i=l , 2)
(24)
then PROSP is solvable. (The outline of the proof) Condition (Ill) in i ) is derived from the fact that any pole of Lci ' which is unstable, may not be contained in the set of fixed modes for satisfying IS. The follOl,ing lemma shows that ii) is equivalent t o i). Suppose that
[Lenma 2]
,
[~:
C 2
* BlQ l B2Q2 0 0
·o 1= n+ql+ q2 o
. (column full rank )
(25)
s. Hara
1198
Condition (Ill) ·in Theorem 3 holds, if and only if (24) is satisfied. (The proof is omitted) 0
(I)
In some spec ial cases, t~o effective necessary and sufficient conditions and a simp l e sufficient condition for PRDSP are derived from Theorems 2 and 3. (The proofs of the se corollaries are omitted.) [Corollary 1 ]
Suppose that the matrix F in (2.a) is represented by F = diag { ~. }
(26)
J
PRDSP is solvable, if and onl y if the condition in Theorem 3 ( i) or ii) ) holds. 0
{{f; ;llH~; ::J[[ 'i]['fll}
then PRDSP is so lvahl e. (The outline of the proof) This theorem is derived from Lemma 8 in the literature by Davison 0 and Gesing( 19 79) and Co roll ary 3. [Remar k 4]
No te that the robust decentralized servo problem is so lvabl e only if m ~ Pi (i= l, 2) but that this is i not required for the so lv ability of PRDSP discussed in this paper (See Example I stated beloK) . Hence, the converse of Theorem 4 is not ge nera ll y true. [Example 1] A decentralized control system having no fixed mode Lp wi th
Cl =
[Corollary 2]
This corollary stated that the ca lculation of the fixed modes, which is rather trouble some, does not required for the solvability check of PRDSP in the case where the number of the control inputs is less th an or equal to that of the control led outputs in each control agent.
p
*BIG l 0
Cl C 2
';;,]
0 I
~ J.
0
0
I ]
* where 6 =1, 8 * =2, 8 * =3 , 2 1 3 is considered. This system does not satisfy the r obust decentralized servo co nditi on (Davison et. a!., 1976, 1979, 198 2), since m (=1) < PI (=2). l There, therefor e , exis t s no decentralized robust se r vo compe nsator satisfying the two robust servo condi tions for any externa l signal . However, in a case where onl v two outputs YIl and Y2 1 are re-
So (8) = I 3 ' SI (e) (27)
= n+P l+P2
C =[ 0 2
under the prescribed partial parameter perturbations 8 (i=I,2,3). In thi s case, (18) has a i unique so lution of
There exist mi xPi matrices C (i=l, 2) such i that
[,'-"
[~
quired to track the reference inputs of step type and ano ther output Yl2 should be regulated at zero
[Coro llar y .3 ]
If condition (I) in Theorem 2 and the following condition: (I I)
(29)
do not contain th e modes \ of F,
Eq. (26) holds~r step functions, but it does not hold for ramp functions. Thi s means that PRDSP may be so l vah le not for step type reference inputs but for ramp type commands when the condition in Th eorem 3 i s not sa ti sfied . Assume that m ~ Pi (i= l, 2) . PRDSP is solvable, if i and only if the condition in Theorem 2 holds. 0
( 17)
(II) •
[Remark .3 ]
Condition ( Ill ) in Theorem is a dual form of Davison's condition, condition ( 11) # in Theorem 4 proposed later, except matrices Q (i=I,2) are ini cluded. The dualit y is due to the configuration of the designed servo system stated before. The matrices Q play the role of se lecting the indisi pensable input classes or ranges to make th e required internal modelS of the reference commands.
~ ({C i }, A*, { B: : ) cC-
[ 9 1 , e 2 ] , S 2 ( e) =
[~
1 -0
1'
since
( V '(O (F)
0
holds, then PRDSP is so l vab l e.
Eq. (27) requires m ~ Pi (i=l, 2) ,si nce the matrices i B~C. have onl\". p.1 co lumns respectivel\·. Therefore, 1 1 .
Thus, "e get q l=I< 2=PI . Q2= 1=I=P 2' and then it is
sho~n
that PRDSP is solvable from Theorem 3.
thi s condition is st r onger than that of
o
'{
l'~:I 'n
SY\THES I S ALGOR I TH~I OF PARTIALLY ROBUST DECE\TRALl:ED SERrO SYSTE~I
(28) ( V\
s :;(F)
whi ch i s regarded as the solvabi lit y condition f or robust centralized se rv o prob l em (Ha ra, 1982 ) Furthermore, it is shOl,n that th e condition for robust decentralized servo prob l em pr oposed by Davison et. al. (19 76 , 1979, 1982) is a sufficient condition for PRDSP as expected.
In this chapter, a svnth esis algorithm of partially robus t decentral i : ed sen·o s,·s tems and the orders of th e designed se rvo compensato r s are investigated based on th e re s ult s obtained i n the previous chapt ers. A synthesis algorithm is proposed in the case Khere the minimal polvnomial of F is represented bv ':' (F)
[Theorem 4]
If the
follo~ing
t~o
conditions hold:
For s t ep and ramp functions, k=l, ) 1=0 and k=2,
(30)
1199
Robust Se rvo Problem for Decentralized Control Systems The following lemma, which
Step 3 :
Let the parameter s of the se r vo compensators Lci' Aci' Bci' Cci ' and Dc i be
gives an equi valent condition of condition (11) in Theorem 2, leads t o a synthe s is algorithm. A .
[Lemma 3) Suppos e that the minimal polynomial ofi'iSrepre se nted by ( 30). Then condition (Il) in Theorem 2 hold s , if and onlv if the following two conditi ons hol d
Bci=[ 0, In "]
Cl
(39)
1
C . = [I Cl
mi
,0)
Dci=[Imi,Oj
( i=I,2 )
Construct stabilizer s L . which stabilize
i)
Pg
["el C,
C2
8 , (8) 8 (8) 2
°
N,
0
0 0
N2
0
].
SI
['Iel e,lel 8,lel] P
g
Cl
0
0
C2
0
0
( V 6 ( 1"1;
the total decentralized servo sy s tem by means of the algorithm proposed by I'o'ang and Davison ( 1973) .
; (31)
i i) There exi s ts \"( 9) =[V (8) T, V (8) T)T (V I 2 satisfying VI (8)
Pg [
V/ 8 )
E VI (8) ml E V (8) m2 2
]
P
r
V l (8)
gL V (8) 2
the designed servo compensat ors Lci in Ste p 3 has no unstable fixed mode.
]
(32)
V' 8 ( 1"1 *
Lastly, a corollary with respect to the ord e rs of the designed servo compensators is presented . [CorolZary 4) Suppose that PRDSP is solvable and that the minimal polynomial of F is given by (30) . If
(
Here
[Remark 5) This algorithm i s useful not only under the c ondition of Theorem 3 but also in the case where the composite system made up of the given plant [ p and
Pg
A(8)_ " I 8,( 8 ) 8 2 (8)] C, 0 0 [ C 0 0 2
=
(n+m +m ) k l 2
(40)
(" "( O(F )
(33) 0
E .= j
I.
0
°klj
holds, then the minimal orders of the servo compensators required in the i-th control agent are P[P,() =n# .
0
)
0 .. , . . . ' 0
1
I. )
1
, (34)
Ij
If (40) holds, then (33) has a uni que (proof) Thus n# are minimal order . solution. 1
o
[Remark 6)
(j=I,2, .... )
(i=I,2)
A( 8 )
(3S.b)
(3S.c)
A numerical example is now presented in order to show that the total order of the designed servo compensators ma y be smaller than that of the conventional one.
(3S.a)
8,(( 8 )
blocks (i=I,2)
C. <-
blocks (i=I,2)
2
the total I: p[P,() ~ (ql+q2)k~ (Pl+P2)k 1=1 order of the compensators designed by the proposed algorithm is less than or equal to those of the servo compensators for the conventional robust decentralized servo systems proposed by Davison et. a!. (1976, 1979, 1982). It is also easil y shown that this inequality generally holds including the case where (40) is not satisfied.
Since
0
[Examp le 2)
We consider a decentralized control s ystem Lp with
A Synthesi s Algorithm o f Partial I,' Robust Decentralized Servo System
~
Obtain a ma trix Q e ) =[v (e) T,V (8) T)T ( V l 2 satisfyin g (33),
Ste p 2
Define matri c e s P,((i =I,2 ) a s
P,
(36)
<-
where d tj
vi (e )= _ g .. (6) P. . j=l
1)
(37)
1)
(gij( 8) are independent scalar functions of 8)
[P1,
( p [P.) = p [P~) <<-
-:J
Bl (8 )= [:], B2 (8)=
o where 8 * =1 and 1
[0
[~]
I)
e*2=2
This system is solvable for an y type of reference command, since the condition of Corollary 3 is satisfied for any " in C+. We now investigate the minimal orders of the designed servo compensators of each control agent in the case where the reference inputs are given by yrl=l (step input ) and Yr2 =0. In this case, the solution of (33) is represented by
and choose a nonsingular matrices W (i=I,2) i satisfy ing P'(W,
A(8)= [-:1
(38)
V (8)=[ I 0 )T , VI (8) O
since F=O, Hl=l, and H =0. 2
=8 1
' V (8) 2
=0
,
Therefore, only one
S. Rara
1200
integrator is required in the fir s t control agent and no integrator needs in the seco nd agent. The t ot a l number of integrators t o be required, which i s equal t o one is less th an two that required in the conve nti onal robu s t servo system proposed by Daviso n e t . al., whose a ll co ntrol agents should have an in tegr a t or . The reduction of the total order of th e des i gned servo compensators is due to the restriction for the classes of the perturbed parameters in the plant and the refference inputs.
1 ) the right hand side of (12)
[TC"(
'01
2) the right hand side of (16) : (same as above) 3) the right hand side of (18)
[socelFd ·
CO!'-lC LlJSJON
El
In thi s paper, some nece ssa r y and /o r s uffici ent conditions for the solvability of th e partiall y r obus t decentral i zed se rvo problem di scussed under the realistic r es tri c ti on for th e classes of the parameter perturbation in the given plant and the r efe r e nce commands have bee n derived. A synthesis a l go rithm of robust de ce ntralized se rvo system has bee n also proposed based on the so l vabi lit y conditi ons. Th e derived so l vab ilit y conditions are weaker and th e orders of the se r vo compensators des igned by th e propos ed a l go rithm are lower than those for th e conve nti ona l robu st dece ntralized control investigated by Davison e t . a l . The procedures of th e seque nti a l synthesis (Davison, Gesing, 1979; Davison, Ozgune r, 1982) may be applied t o th e proposed method under an assumption of th e s tabili za bility by th e gain out put feedback in each control agent. The author wishes to thank Prof. K.Furuta of Tok yo In s titut e of Technology fo r his constant encouragemen t durin g thi s work .
( 43)
E2
'0 1 T T NI ' NZ ] in (31) and (33)
4) the matrix [ 0 T
T
NdO
(44)
NT dZ
Ndl
( 45)
where
r E:~d
N..{.
(i=0,1,2)
l'
(46)
E F: kd- l i d
and kd is the order of the minimal polynomial of F . d CASE 2 :
case of containing direct parts
A system with direct parts of u. to y . , whose out1 put equation in each agent is wfitten by (47)
REF ERENCES
is considered.
flengtsso n , G. (1977). Outpu t regulation and i nt erna l mode l s -a frequency domain approach. Automatica , 13, 333 - 345. Da vi so n, E. J. (1976). The robu s t decentralized control of a ge ne r a l se rvomec hanism problem. IEEE Tr ans. Autom. Co ntrol, AC-2l, 14-24. Davison, E. J., and Gesing, W. (1979). Sequential s t abi lit y and op tim iza tion of large scale decentralized sys tems. Automatica, 15, 307324 . Daviso n , E. J., an d Ozguner, D. (1982). Synthesis of the decentralized r obust servomechanism prob le m usin g lo cal models. IEEE Tran s. Autom. Con trol, AC-27, 583-600. Hara,5.(f982). So l vabi l i t y of servo problem for pa rtial parame t er pe rturbati ons. Tran s . of SICE, 18, 879 - 885 (in Japanese). lI'ang,s,-and Davis01l, E. J. (1973). On the s tabilization of dece ntra li ze d control systems. IE EE Trans. Autom. Con trol, AC-18,473-485.
Sen'o Problem for Disturbance Input s
We consi de r a system (41. a) Y. . 1
C.x 1
+
E.d 1
( i = 1,2)
(2,2) (3,3) 2) Eq . (23) [Cl 0 [C 2 0
OJ
~
0]
[Cl Dl o .1 [C 2 0 D2J
3) Eq . (29)
[:;Hl
~
[:J]
4) Eq. (31) and Eq. (33)
[:~l (same as above)
[Remark 7 ] 1) When the plant have three or more control agents, the results obtained in this paper can be extended by the expansion of the corresponding matri ce s.
APPE:-
1) (2,2) and (3,3) element of the matrices in the left hand sides of (18), (25), and (27)
(4 1.b)
where d is an nd-dimensional di s turbance input vecto r which is ge ne rat ed by a linea r fre e sys tem expressed b:' (42) Th e r esults for dis turb ance input s can be obtained by replacing the following equa tion s in the theorems, corollaries, and l emma s proposed in eac h chapter.
2) Note that the servo problem or the servo synthesis for reference commands and for disturbance inputs can be simultaineouslY treated . The more lower order servo compensators may be designed by gathering these two external signsls .
©
l\u
11 1111"-:,11\ .
11-".\(
(Jlh
I
Ill·rrlll,!!
\\"(orld (tlrr .~H·"
I~"""l
ROBUST SERVO PROBLEM FOR DECENTRALIZED CONTROL SYSTEMS WITH PARTIAL PARAMETER PERTURBATIONS S. H ara
Ahstract. This paper is concerned with the partially robust servo problem for decentrali:ed contro l svs tems with partia l parameter perturbations in the case where the outputs of each con trol agent are required to track different types of reference commands ~ithout steady-s tate errors. Necessary and / or sufficient conditions for the sOlvabilitv are derived. These lead to the fact that conventional solvabi l ity condition for th e robust decentralized servo problem presented by Davison et. al. is only a sufficient condition of this problem. A synthesis procedure of decentralized servo svs t em. ~ith matrix opera tions is proposed based on these solvabilit y conditions. It is also shown that the total order of the designed servo compensators may be sma l ler than that of the conve ntiona l one. Kev~ords . Se rvomecha nisms; Decentralized contro l systems; Large-scale systems; System anulvsis; Contro l sys tem synthesis; Parameter perturbation
I\TROnUCTIO,<
to the fact t hat the conventional solvability condition for t he robust decentra l ized servo problem investigated by Davison et.al. (1976, 1979, 1982) is only a sufficient condition of this problem. In Chapter 4, a synthesis procedure of partially robust decentralized servo sys tems is proposed based on these solvability conditio ns. I t is a l so shown that the to t al order of the designed servo compensators may be smal l er t han that of the conventional one.
Recently, much attention has heen paid to decentralized control systems with the enlargeness of plants to be controlled and the advance of computers. Both analysis and s"nthesis for decentra liced control systems have been investigated by many researchers. In connection with the servo problem Ilavison et. a!. (1976, 1979, 1982) have proposed some useful synthesis algorithms for rohust decentralized servo controllers under the follo~ing assumptions: 1) All the parameters of the plant vary independ e ntly; 2) Al l the externa l signals such as refe r ence commands and disturbance inputs are generated bv sys tems with the same modes . These results are effective for decentralized systems with parameter perturbations, but they have sho rtcomings in that s tr ong restrictions are imposed on the plant for the robust servo problem hei ng solvable and that the order s of the compensa tor s designed are necessarily large due to above unrealistic assumptions. In practice, however, ~' e oftE'n encounter the following cases: 1) Particular invariant characteristics such as integra l characteristics in d.c . motors exist in the plant. This means that the plant parameters are not all perturhed, i.e. , some para~eters are fixed at zero, one, etc. ; 2) All the external signals do not have the same modes . For example in crane svstems, the position of the cart has to track the reference input of s tep type without steady -state error but the angle of the rope is onlv re quired to be regulated at zero .
Notation: alA), p[A ], and A+ denote the set of the eigen values, t he r a nk , and the seude i nverse of a constant ma t rix A, respec t ively. For a ma t rix A(e) which is a f unction of ~erturbed parame t ers e, A* and Pg [A(e) ] denote A(e ) and t he ge neric rank, which is equa l to the maximam rank, of A(e), respectively. C+ and C- represent the closed right half complex plane and the open left half complex plane, respectivel y. PROBLE~l
STATE~lENT
I n this paper, we consider a decent r alized control system wi t h partial parameter perturbat i ons [peAce), {B ( e)}, {C })
i
i
x = A( 6) x + Bl (e)u
l
+ B2 (8) u 2 (i = 1,2)
Yi= Cix
This paper , therefore , concerns Idth the ro bust sen'O problem for decentral i zed control systems with pa rtial parameter perturbations in the case ~here the ou tput s of each control agent are required t o track different tvpes of reference commands without steady-state errors. In order to avoid the complexit v of the description this paper only deals with the case where 1) the external signals contain no disturbance inputs, :) thE' direct pa r ts of the inputs to the outputs do not exist, 3) there are only t~o control agents . Results obtained here can be easilv extendE'd t o the general case. (See Appendix.)
(I. a)
(1, b)
where xERn is a state vector, UiERmi and YiERPi (i= 1,2) are an input vector and an output vector of i - th control agent respectively, and eER r is an unkno~n and unmeasurab l e parameter vector with nominal value e *. The reference inputs Yri (i= I, 2) are given by the outputs of a linear f r ee sys t em [ rtF, {Hi }) Fz
(a(F) c C+)
(2.a)
(i= 1,2)
(2. b)
~here ZERnZ, "r.ERPi (i=I,2). . 1
The error vectors, I;hich are equal to the measura ble vec t ors, of each control agent ar e defined by
The organization of this pape r is as follows: In Chapte r 2, a partially robust decentralized servo problem (denoted by PRDSP) is defined . Necessary and/or sufficient conditions for the solvability are derived in Ch3ptcr 3. These condi t ions lead
(i = 1,2)
He r e, i t is assumed that a pair (A*, [B ,B ]) is l 2 1195
(3)
s.
1196
rCI],A)*
controllable and that pairs ( ic l
z (0), '9' -xI (0), "1 -x (O), an d "I q. [ ~.. f:*) . 2 Especially, PRDSP is so l vable by static compe nsators, if there exist a positive number [ and Zi ) ( V x(O,
and
2
both observable. [Remark 1) Lr(F,{Hi)) is a general form of reference commands, si nce polynomial and sinusoidal tYpe reference commands which are often appeared in the practical servo synt hesis can be expressed by this sys tem . For example, I ) Step: F=O H.= I 1
2) Ramp:
F= [ : :]
3) Sinusoidal: F= [ 0
W
Ha r a I(
wit ~i(the order of f i)=o ; i=I,2 such that the above two conditions hold. IS requires the stability of the designed closed loop system in the case where all the reference commands are equa l to zero. OR means that al l th e outputs track each reference input I,ithout steadystate errors . La stlv, the set of fixed modes, which is an important concept for investigating the stabi lity of decen trali zed systems, is stated .
1
-w 0 [De finiti on 2 ) (Wang and Davison, 19 73)
We now consi der a decentralized servo system with dynamic compensators having constant coefficient l: i
(Ai ,Bi ,c\ ,D i ) A.x.
+
Ri Yei
(4. a)
u.
C.x.
+
DiYei
(4 . b)
1
1
1
1
where ~ . E Rni
p -
The composite sys t em made up of Lp and Li (i=I,2) is des cribed by =
~(~)~
(S . a) (i =1,2) ,
Y i = Ci x
and
(S.b)
_
1
=[
-B (8)C
2
-BIC I
Al
0
-B 2C2
0
/\2
C.
0
0
1
= {
s ! 11 9 - e* 11 < [ }
) . (6. b)
(7)
i:ed servo pr oblem (PRDSP) is solvab le at e=e* , if there exis t a positive number [ and decentrali:ed dynamic compensators ~i(~i,Bi,Ci,6i) ; i=I,2 such
v
ei (t)
=
0
; i==i,2
( ~8E
0.;)
(12)
(The outline of the proof) Conditions (I) and (II) correspond to the TS and OR conditions, respectively.
0
Condition ( I ) is a necessary and sufficient condition for stabilit,· by gain output feedback in each agent. Condition (11) i s that of a ce ntral i:ed sy s tem having an internal model of reference comTherefor e , PRDSP b,· s t a tic mands. (Hara , 1982) compensators is not different from the case of the ce ntrali: ed servo problem by gain output feedback . [Rema rk 2 ) that the property of internal stability is well-posed for parameter perturbation, s o that if the condition (8) holds only at the no minal value ~ = 9 * then the intern al stability condition i s al s o satisfied. Condition (T ) in Theorem I thu s consists of a condition a t the nominal value ~ * . Thi s i s true in the fOllOldng theorem s , lemmas , and corollarie s . ~ote
(8)
(9)
lik e th os e inv e stigated h,· Davison et. a1. ( 1976, 19 79, 1982). HOI,ever, the po s ition s are reversed, i.e., Lci are follol,ed b,· =si in Fig.1 but :: si are
(11) Output Regulation (OR)
ZiT'"!
['COlHI F
the so l vabi lit y condition s by dynamic compensators is discu s sed. Without lo s s of generality, a decentralized servo system in Fig.1 is considered he r eaft er. Th e control l er con s i s ts of s ervo compensators =ci for OR and sta b ili:ers =si for IS
that the follol,ing tl,O conditions hold (I) Int ernal Stability (IS)
t~oo
(11)
H2 .
[De fini t i on 1 ) When a reference command is generated by a system ! r(F, {Hi }) for a decentralized control system : p (A( e), {Bi( e ) i , {Ci 1 ), the partially robust decentral-
(y S€:::
2 * L B. K. C. ) c:. Ci= I I I I
'(O l ~~ T( 8)
, (6. a)
A partially robust servo problem for decentralized con trol systems is thus defined as follOl,s :
O(A( e )) C C-
+
l1
2
This composite system is required to retain two servo properties, namely internal stability (IS) and output regulation(OR) , for all parameters 8 contained in th e [ -neighbourhood at 8 = e * r epresented by
,,* .. [
*
(I I) There exist a posi tive number [ and an nxn z matrix T( 6) such that
_
A(8)- L B. (8)D.C. , -B (8)C I I i=1 1 1 1
C.
(10)
[Theor em 1 ) PRDSP is solvable by static compensators, if and only if the following two conditions hold (I) There exist mi xPi matrices Ki (i= I, 2) such that
T
2
A( 8)=
. ., * f B.I\ . C.) i=lll1
First, a necessary and sufficie nt solvability condition for PRDSP hy static compensators is proposed.
(A x
+
SOLVAR ILITY CONDITIONS rOR PRDSP
where x::;
*
{S;},{C ; ) .
No te that the in puts and
the outputs of these systems are measurable outputs and control inputs of each agent in L , r espective l y.
~
(A
i
( i =I,2).
1
n
{K. }
is said to be the set of fixed modes for Lp(A * ,
~
1
1
1
x. 1
* !' ( {C · } ,A , {B*I· }) =
~ext,
folloh·ed b,· ::ci in Da\·lson ' s controll e rs.
The
Robust Servo Problem for Decentralized Control Systems
11 97
contains the para meters of Lci to be designed. Thus, th e necessary and / or sufficient conditions represented only by the parameters of the given plant [ are proposed . p [The orem 2)
If PROSP is solvable, then the two conditions hold: (I) fI({C }, / , {S:}) i
Fig . l
If the servo compensators Lci (Aci,Bci,Cci,Oci) are represented by
x. Cl
A .x.
+
B
.u .
(13. a)
u.
C .x .
+
0 .u .
(13.b)
Cl Cl
Cl Cl
1
Cl Cl Cl Cl
where x . £Rnci and u .£Rmci Cl
Cl
then the composite
e-
(1 7)
(11) There exist a positive number an nxn z matrix SO(9) and mixn
Block diagram of decentralized servo sys tem
configuration of the servo system in Fig . l is same as that of the centralized servo system treated by Bengtsson(19 77) and Hara(1982). It can be expected that the weaker solvability conditions and the lower orders of the compensators are obtained by this method, since the invariant characteristics such as integral characteristics contained in the given plant may be available for the internal model in this configuration .
c
follo~ing
z
and matri-
ces Si (9) (i=I,2) such that A( e) Bl (e) B2(e) Cl 0 0
1
[SO( 9) -j _ [SO( 9) FjSI (9) HI
[ C 0 0 S2( 9) H2 2 (The outline of the proof) IS condition requires condition (I). The fact that condition (Il) is equivalent to condition ( 11 ) in Lemma 1 which must be satisfied for OR. 0 When condition (11) in Theorem 2 is satisfied, the solutions Si (e) (i=1,2) can be written by hi
d \
(9)
L f . . (9)Q ..
j=l
IJ
(19)
1)
and define qi as
system r(L ,{L . }) is written by p
Cl
(20)
(l4.a) (l4 . b)
where f (9) are independent scalar function s of 9 , ij and (21)
[Q·l'Q·2' .... · 'Q·1 h) 1 1 i
where x
=
(9)1) (i=I,2). (22) ihi We now get two equivalent sufficient conditions for PROSP by means of the constant matrices ~ . [fil (e)I,· ··· .,f
T
x
and
[Theorem J)
i) If conditions (I) and (11) in Theorem 2 and (Ill) There exist mixqi matric es Qi (i =1,2) such
,~[:: ;h;l}R'r:;:: 'e!~:grj :": ; :~}
(23)
do not contain the modes A of F A necessary and sufficient condition for PROSP is represented by the composite system L in the same manner of Theorem 1. (The proof is omitted.)
[Lemma 1) PROSP is solvable, if and only if there exist Lci (i=I,2) such that the following two condi-
or the equivalent ii) If condition (I) in Theorem
,
p[~)
tions hold : (I)
fI({e),A, {(})
c
e-
(IS)
(Il) There exist a positive number £ and an (n+n
+nc2)Xnz matrix fe e) such that [ A(e)
~~
1
T( 6)
~
[':;)F 1
C V 6 Ell;)
cl
(16)
0 This lemma, however, may not be useful for the design of Lci' because the condition of this lemma
holds and
(Il ) Condition (Il) in Theorem 2 holds with
qi
~
Pi
(i=l , 2)
(24)
then PROSP is solvable. (The outline of the proof) Condition (Ill) in i ) is derived from the fact that any pole of Lci ' which is unstable, may not be contained in the set of fixed modes for satisfying IS. The follOl,ing lemma shows that ii) is equivalent t o i). Suppose that
[Lenma 2]
,
[~:
C 2
* BlQ l B2Q2 0 0
·o 1= n+ql+ q2 o
. (column full rank )
(25)
s. Hara
1198
Condition (Ill) ·in Theorem 3 holds, if and only if (24) is satisfied. (The proof is omitted) 0
(I)
In some spec ial cases, t~o effective necessary and sufficient conditions and a simp l e sufficient condition for PRDSP are derived from Theorems 2 and 3. (The proofs of the se corollaries are omitted.) [Corollary 1 ]
Suppose that the matrix F in (2.a) is represented by F = diag { ~. }
(26)
J
PRDSP is solvable, if and onl y if the condition in Theorem 3 ( i) or ii) ) holds. 0
{{f; ;llH~; ::J[[ 'i]['fll}
then PRDSP is so lvahl e. (The outline of the proof) This theorem is derived from Lemma 8 in the literature by Davison 0 and Gesing( 19 79) and Co roll ary 3. [Remar k 4]
No te that the robust decentralized servo problem is so lvabl e only if m ~ Pi (i= l, 2) but that this is i not required for the so lv ability of PRDSP discussed in this paper (See Example I stated beloK) . Hence, the converse of Theorem 4 is not ge nera ll y true. [Example 1] A decentralized control system having no fixed mode Lp wi th
Cl =
[Corollary 2]
This corollary stated that the ca lculation of the fixed modes, which is rather trouble some, does not required for the solvability check of PRDSP in the case where the number of the control inputs is less th an or equal to that of the control led outputs in each control agent.
p
*BIG l 0
Cl C 2
';;,]
0 I
~ J.
0
0
I ]
* where 6 =1, 8 * =2, 8 * =3 , 2 1 3 is considered. This system does not satisfy the r obust decentralized servo co nditi on (Davison et. a!., 1976, 1979, 198 2), since m (=1) < PI (=2). l There, therefor e , exis t s no decentralized robust se r vo compe nsator satisfying the two robust servo condi tions for any externa l signal . However, in a case where onl v two outputs YIl and Y2 1 are re-
So (8) = I 3 ' SI (e) (27)
= n+P l+P2
C =[ 0 2
under the prescribed partial parameter perturbations 8 (i=I,2,3). In thi s case, (18) has a i unique so lution of
There exist mi xPi matrices C (i=l, 2) such i that
[,'-"
[~
quired to track the reference inputs of step type and ano ther output Yl2 should be regulated at zero
[Coro llar y .3 ]
If condition (I) in Theorem 2 and the following condition: (I I)
(29)
do not contain th e modes \ of F,
Eq. (26) holds~r step functions, but it does not hold for ramp functions. Thi s means that PRDSP may be so l vah le not for step type reference inputs but for ramp type commands when the condition in Th eorem 3 i s not sa ti sfied . Assume that m ~ Pi (i= l, 2) . PRDSP is solvable, if i and only if the condition in Theorem 2 holds. 0
( 17)
(II) •
[Remark .3 ]
Condition ( Ill ) in Theorem is a dual form of Davison's condition, condition ( 11) # in Theorem 4 proposed later, except matrices Q (i=I,2) are ini cluded. The dualit y is due to the configuration of the designed servo system stated before. The matrices Q play the role of se lecting the indisi pensable input classes or ranges to make th e required internal modelS of the reference commands.
~ ({C i }, A*, { B: : ) cC-
[ 9 1 , e 2 ] , S 2 ( e) =
[~
1 -0
1'
since
( V '(O (F)
0
holds, then PRDSP is so l vab l e.
Eq. (27) requires m ~ Pi (i=l, 2) ,si nce the matrices i B~C. have onl\". p.1 co lumns respectivel\·. Therefore, 1 1 .
Thus, "e get q l=I< 2=PI . Q2= 1=I=P 2' and then it is
sho~n
that PRDSP is solvable from Theorem 3.
thi s condition is st r onger than that of
o
'{
l'~:I 'n
SY\THES I S ALGOR I TH~I OF PARTIALLY ROBUST DECE\TRALl:ED SERrO SYSTE~I
(28) ( V\
s :;(F)
whi ch i s regarded as the solvabi lit y condition f or robust centralized se rv o prob l em (Ha ra, 1982 ) Furthermore, it is shOl,n that th e condition for robust decentralized servo prob l em pr oposed by Davison et. al. (19 76 , 1979, 1982) is a sufficient condition for PRDSP as expected.
In this chapter, a svnth esis algorithm of partially robus t decentral i : ed sen·o s,·s tems and the orders of th e designed se rvo compensato r s are investigated based on th e re s ult s obtained i n the previous chapt ers. A synthesis algorithm is proposed in the case Khere the minimal polvnomial of F is represented bv ':' (F)
[Theorem 4]
If the
follo~ing
t~o
conditions hold:
For s t ep and ramp functions, k=l, ) 1=0 and k=2,
(30)
1199
Robust Se rvo Problem for Decentralized Control Systems The following lemma, which
Step 3 :
Let the parameter s of the se r vo compensators Lci' Aci' Bci' Cci ' and Dc i be
gives an equi valent condition of condition (11) in Theorem 2, leads t o a synthe s is algorithm. A .
[Lemma 3) Suppos e that the minimal polynomial ofi'iSrepre se nted by ( 30). Then condition (Il) in Theorem 2 hold s , if and onlv if the following two conditi ons hol d
Bci=[ 0, In "]
Cl
(39)
1
C . = [I Cl
mi
,0)
Dci=[Imi,Oj
( i=I,2 )
Construct stabilizer s L . which stabilize
i)
Pg
["el C,
C2
8 , (8) 8 (8) 2
°
N,
0
0 0
N2
0
].
SI
['Iel e,lel 8,lel] P
g
Cl
0
0
C2
0
0
( V 6 ( 1"1;
the total decentralized servo sy s tem by means of the algorithm proposed by I'o'ang and Davison ( 1973) .
; (31)
i i) There exi s ts \"( 9) =[V (8) T, V (8) T)T (V I 2 satisfying VI (8)
Pg [
V/ 8 )
E VI (8) ml E V (8) m2 2
]
P
r
V l (8)
gL V (8) 2
the designed servo compensat ors Lci in Ste p 3 has no unstable fixed mode.
]
(32)
V' 8 ( 1"1 *
Lastly, a corollary with respect to the ord e rs of the designed servo compensators is presented . [CorolZary 4) Suppose that PRDSP is solvable and that the minimal polynomial of F is given by (30) . If
(
Here
[Remark 5) This algorithm i s useful not only under the c ondition of Theorem 3 but also in the case where the composite system made up of the given plant [ p and
Pg
A(8)_ " I 8,( 8 ) 8 2 (8)] C, 0 0 [ C 0 0 2
=
(n+m +m ) k l 2
(40)
(" "( O(F )
(33) 0
E .= j
I.
0
°klj
holds, then the minimal orders of the servo compensators required in the i-th control agent are P[P,() =n# .
0
)
0 .. , . . . ' 0
1
I. )
1
, (34)
Ij
If (40) holds, then (33) has a uni que (proof) Thus n# are minimal order . solution. 1
o
[Remark 6)
(j=I,2, .... )
(i=I,2)
A( 8 )
(3S.b)
(3S.c)
A numerical example is now presented in order to show that the total order of the designed servo compensators ma y be smaller than that of the conventional one.
(3S.a)
8,(( 8 )
blocks (i=I,2)
C. <-
blocks (i=I,2)
2
the total I: p[P,() ~ (ql+q2)k~ (Pl+P2)k 1=1 order of the compensators designed by the proposed algorithm is less than or equal to those of the servo compensators for the conventional robust decentralized servo systems proposed by Davison et. a!. (1976, 1979, 1982). It is also easil y shown that this inequality generally holds including the case where (40) is not satisfied.
Since
0
[Examp le 2)
We consider a decentralized control s ystem Lp with
A Synthesi s Algorithm o f Partial I,' Robust Decentralized Servo System
~
Obtain a ma trix Q e ) =[v (e) T,V (8) T)T ( V l 2 satisfyin g (33),
Ste p 2
Define matri c e s P,((i =I,2 ) a s
P,
(36)
<-
where d tj
vi (e )= _ g .. (6) P. . j=l
1)
(37)
1)
(gij( 8) are independent scalar functions of 8)
[P1,
( p [P.) = p [P~) <<-
-:J
Bl (8 )= [:], B2 (8)=
o where 8 * =1 and 1
[0
[~]
I)
e*2=2
This system is solvable for an y type of reference command, since the condition of Corollary 3 is satisfied for any " in C+. We now investigate the minimal orders of the designed servo compensators of each control agent in the case where the reference inputs are given by yrl=l (step input ) and Yr2 =0. In this case, the solution of (33) is represented by
and choose a nonsingular matrices W (i=I,2) i satisfy ing P'(W,
A(8)= [-:1
(38)
V (8)=[ I 0 )T , VI (8) O
since F=O, Hl=l, and H =0. 2
=8 1
' V (8) 2
=0
,
Therefore, only one
S. Rara
1200
integrator is required in the fir s t control agent and no integrator needs in the seco nd agent. The t ot a l number of integrators t o be required, which i s equal t o one is less th an two that required in the conve nti onal robu s t servo system proposed by Daviso n e t . al., whose a ll co ntrol agents should have an in tegr a t or . The reduction of the total order of th e des i gned servo compensators is due to the restriction for the classes of the perturbed parameters in the plant and the refference inputs.
1 ) the right hand side of (12)
[TC"(
'01
2) the right hand side of (16) : (same as above) 3) the right hand side of (18)
[socelFd ·
CO!'-lC LlJSJON
El
In thi s paper, some nece ssa r y and /o r s uffici ent conditions for the solvability of th e partiall y r obus t decentral i zed se rvo problem di scussed under the realistic r es tri c ti on for th e classes of the parameter perturbation in the given plant and the r efe r e nce commands have bee n derived. A synthesis a l go rithm of robust de ce ntralized se rvo system has bee n also proposed based on the so l vabi lit y conditi ons. Th e derived so l vab ilit y conditions are weaker and th e orders of the se r vo compensators des igned by th e propos ed a l go rithm are lower than those for th e conve nti ona l robu st dece ntralized control investigated by Davison e t . a l . The procedures of th e seque nti a l synthesis (Davison, Gesing, 1979; Davison, Ozgune r, 1982) may be applied t o th e proposed method under an assumption of th e s tabili za bility by th e gain out put feedback in each control agent. The author wishes to thank Prof. K.Furuta of Tok yo In s titut e of Technology fo r his constant encouragemen t durin g thi s work .
( 43)
E2
'0 1 T T NI ' NZ ] in (31) and (33)
4) the matrix [ 0 T
T
NdO
(44)
NT dZ
Ndl
( 45)
where
r E:~d
N..{.
(i=0,1,2)
l'
(46)
E F: kd- l i d
and kd is the order of the minimal polynomial of F . d CASE 2 :
case of containing direct parts
A system with direct parts of u. to y . , whose out1 put equation in each agent is wfitten by (47)
REF ERENCES
is considered.
flengtsso n , G. (1977). Outpu t regulation and i nt erna l mode l s -a frequency domain approach. Automatica , 13, 333 - 345. Da vi so n, E. J. (1976). The robu s t decentralized control of a ge ne r a l se rvomec hanism problem. IEEE Tr ans. Autom. Co ntrol, AC-2l, 14-24. Davison, E. J., and Gesing, W. (1979). Sequential s t abi lit y and op tim iza tion of large scale decentralized sys tems. Automatica, 15, 307324 . Daviso n , E. J., an d Ozguner, D. (1982). Synthesis of the decentralized r obust servomechanism prob le m usin g lo cal models. IEEE Tran s. Autom. Con trol, AC-27, 583-600. Hara,5.(f982). So l vabi l i t y of servo problem for pa rtial parame t er pe rturbati ons. Tran s . of SICE, 18, 879 - 885 (in Japanese). lI'ang,s,-and Davis01l, E. J. (1973). On the s tabilization of dece ntra li ze d control systems. IE EE Trans. Autom. Con trol, AC-18,473-485.
Sen'o Problem for Disturbance Input s
We consi de r a system (41. a) Y. . 1
C.x 1
+
E.d 1
( i = 1,2)
(2,2) (3,3) 2) Eq . (23) [Cl 0 [C 2 0
OJ
~
0]
[Cl Dl o .1 [C 2 0 D2J
3) Eq . (29)
[:;Hl
~
[:J]
4) Eq. (31) and Eq. (33)
[:~l (same as above)
[Remark 7 ] 1) When the plant have three or more control agents, the results obtained in this paper can be extended by the expansion of the corresponding matri ce s.
APPE:-
1) (2,2) and (3,3) element of the matrices in the left hand sides of (18), (25), and (27)
(4 1.b)
where d is an nd-dimensional di s turbance input vecto r which is ge ne rat ed by a linea r fre e sys tem expressed b:' (42) Th e r esults for dis turb ance input s can be obtained by replacing the following equa tion s in the theorems, corollaries, and l emma s proposed in eac h chapter.
2) Note that the servo problem or the servo synthesis for reference commands and for disturbance inputs can be simultaineouslY treated . The more lower order servo compensators may be designed by gathering these two external signsls .