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Algorithms for Large Scale System Control Design
ALGORITHMS FOR LARGE SCALE SYSTEM CONTROL DESIGN L. Bakule Institute of Information Theory and Automation, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
Abstract.The paper deals with the interactive cOTIp ~ter aided control design alsori tlms for the continuous linear dynamic systems . The decentralized pole assiGr~1'1ent problerll for tne strongly connected suhsystems using the dynamic compensation a:::Jo. the t'No-stage optinal control desi gn al~orithD reliable under tbe structural pe rturbations of the interconnections are cOl".sidered.:rhe iilteractioll by the COTJputetio n concerns the system strl1cture Bnd tbe data specification, the eigenvalues specification and the Graphical representatio:: of the desired veriables as e step responses. The case study of the power system mode l is included. Ee;{\'lords. AIcori tl1nlS for digital computer control; interactive compllter aided desien, large-scale systems; mu l tivar:!_ab l e control; opti1:lal control. I N':"CP..ODTJ CT ION
There i s increasing i n terest in tbe large-scale sys tem Llethodolog::r . A large-scale system may be considered as an interconnection of the s u bs~rste!:1s. ~he co~plexitv of the svstem is nain ly dete~mined - by t ~ ~ in~erconnections . A desirable feature is the separation at the subsyste~s a n ~ the i~te~con nection level. ':'he :::ost 13:i.nple approach is to ne~lect the interconnections and treat t~e lower order s ubsystems only. J uch a n approach works well if there is little interaction occ uring between controllers. Let us deal VIi th two procedures for the control design of t he large-sc/)le linear d;,rnaT!1ic s ysteDs, \'lhe re th.e i ::1terac tior.s infl1..1el:ce Ol·, t h0 s:',Tste r,~ activity. ~hc dcca n tr3li3ed polo ass:i_cr.!lellt proced :.~ ro cleels ;-:i th t~l.C c1:'nan~,c co:-.rpel~satio :: desiGn for the controllers so that t ~ e closed-loop eic;envalues 2~e prescribe d for the k -cha::1nel , j ointl~ co n trollable, jointly observa ble, si;ron~ly connected s~rsteY.ls [ 5]. :;"11e co nce ~t of t~~e cO:',lplcte s:rster.: is 1.' sed to :Ja l:e the systen bot~ co ntrollable and observa b le throl'3> a s inJle c::a~c -,-, e I b:r t:'-~e applicatioE of t~le :".oi:d:,r:1a:-:ic dece :1tralized feedbac :: to 011 c:Oan':e1s . Then t he Bra sc ~ - ?earso ~ procedure [4Q ca ~
be
't~sed
to construct t :J e
The two -stage optimal control procedure is s upplied by the power system Dodel control del3iGn example.
JJet us fori:iulate the froblern and the l-'ro b J.elii :2 . Problem 1. The system 3 is described as l:iJleer k-ch arcnel system of a form
',Jr=Ax+t 3.u., i= 1 1. 1. ~Ti = Cix, ., WDe r~.
xl:
';'!D \1.
i=1, ••• ,k~2, .
(resp.
11
1li.€ i
(1)
rH.
~
; resp.
y.£R ~ ) is t~e sta te (resp . the co~ ~
trol, resp. t )1 0 ont p,; t) vector. AE. f{n x n , I\€ Rn x nli , C € R !J i x n i are the consta~t matrice s describinG the systefl d;pa mi!:: s. Le t us suppose that S is a joint ly co~tro lla ble , j ointly obs ervable,i .e.
d:n.1 a l~-:i c
lo ca l control resulti ~G in a Cl0BGdJ_oop systen \',i t1'2 prescribed spectr1l!n.
S.F.(,.('. - M*
The two-staGe optimal co n trol desi~n proc ed1.1 re reliable under the str'.1ctl1 ral perturbati ons.Such perturbation C9n resul"; in i :::.stabj.li ty of the S;,TStem ['7J. The fi_rst stage deals \':1 to the local control desien, the second stage deals with the cont rol design minimizing the interconnec tion effect. Its i n fluence o~ the cost criterion is eva l119ted by t?J.e CN1PlltillG of the Sllboptil'!1ali ty index (7).
345
346
L. Bakule
the triple (e,A,B), C = col (C 1 ' ••• ••• ,c1,),:8 ~ ~O\" (B1"",l):) is cor.Jplete15 ,8J.(.) means transpose (.). decentralized pole assignment p!'oblen is to find the dynam:Lc local controllers of c form
~'he
.
zi
=
Sizi + RiYi'
Uin~ Qizi + FiYi' Z~E.TI.
i
= 1, ••• ,k (2)
so thet the closed loop spect.rum of the system (1),(2) is freely assiGnable. Note that the nondynamic loc::ll control has a for]"! l
1.\
= FiYi
min ,T
(4)
U
s\.lbject to
,
' E T) n.x n. -.., ".,·,m. x r.J. n >0 ,./; le l l,hi,,"n l l , ·oi- ,
~
n.
ts both cOY'.trollable and observable t~lrOl.lgh a si::Jgle channel denoted .;i. '1'he11 the dynanic conpensator can De applied to construct a control closedloop systen with prescribed spectrum [~. Theorem 1 (resp. Corollary 1) SOlve this problem [SJ.I'he triple (C,A,B) is complete if (A,B),(A,C) is controllable, observable[s]. n
,
Dj
.
Ri>O; XiE:R l i, resp. uiE:H -) lS the state vector (resp. the control).'l'he system dynamics is described by the constant Elstrices Al' ,B-;... ,A; -;. I,et us suppose k peirs (Ai,n i ) cont!'olable.
n+,.
4C
uj.n
11 A
BFII ,
}3
1
Let us suppose the control D = 1} + 1.11 •
(7)
;3CJIilJIn OH - ALGOEITIIIVI3
I ·et ,'S fo:.~r.;uJ.[lte t~1 e alsori tl1ms on the basis of the knoqn res ults. The solution consists of the solution of Problem 1 (resp. Problem 2) as the Algorithm 1 (resp. Algorit hm 2). The basic approacb to the Problen 1 solution is to apply first local nondynamic controls (J) so tllSt the closed loop system
'"
IIAII~II>B·tl·r.;·II;A '[' l l l c
4)
=
=A +lB.H.r: - - l l.• If it is satisfied Tgo~to 4) else go to end). If j = k-1 go to 5) else go to 3) •
i£(1, ••• ,k} A = A ,n = B., C = C .• c 'it' a-
5) a) Choose
6)
- Q U~ G U~ 1 ' • • • , Al ), = c1i8ij ( B , ... , I\J. (
tJn",/l.. =1
where ~o (resp. ~c) denotes the observability index (resp. the controllability index) [4,5]. Algorithm 1. 1) a) Give A,Bi,Ci,~i,j=O,xo = x(t o ) 2) Test ths joint controllability, joint observability (C,A,]), .£ = (C1"",Ck)'~ =~TIi • If it is not satisfied go to end) • J) j = j + 1. Test the completness (Cj,Ac,B j ) using the pseudorandom generator. So that
(6)
·..,here J,. l\..
.
K.!: ,a 0 ).'f3i
the desired set, cenerally complex conjugate pairs, of the eigenvalues A = [ .A~, ... ,).n+1-1; t- min f ~Ol ale},
~<.l
I,et us find the control optimizing ooth the stlbsystems Beti vi ty and the i nfltiGl1ce of tbe L1terconnections. ']'hei:: find F
.
Denote: I.AI - AI =.[ ).1 ~i the · desired ,-0 characteristic polynomial
(3)
resulting from (2) for n i = O. :Get us formllate the Problem 2 now. Problem 2. Let us ~ind the control fi for the cost
VI.~lGre
(8)
Test the A-cyclic property. If it is not cyclic go to 7) else go to 8). Using the pseugo-random genera.7) tor geDerete H 52 that U,,'..II=IIBHclI, A:=A+BHC; c.;o to 6). G) a) Choose c' = d' C so that (A,c') is ohservable. 9) Compute a column vector :L. I = [L i]
i= 1, ••• , n;
- d'A i - 1 n iri J d'A n - k - 1B·, 1";~+'-~ k -
10 )
ka f - -
Compute q. 11)8) Give A. 12) Solve equation Pe = f, r= m~, ". w~lere e (resp.f) is q+r(q+1)
Large scale system control design
(resp. n+q) column vector, F=[Pij]; Pj=col (P1j"" ,Pn+q, j ); Pj=col( O, ••• ,O,p .'j =1, ,(11-1' ••• , «()' 0, ••• , 0) , j £ [1 , ••• , Pj=col (0 , ••• , 0 , Pjj=L 11 ,L21""
qJ ;
te-time problem reletion [1J. I.et us ~ l rther suppose IBI = m. Algorj. thm 2. 1)8) Give ,\,Bi,Aii,Qi,Ri,i"i,xo=x(to)' tJ
2)
3)
,nJ
fi= ;1q_i_n,iE [n+1, ••• ,n+ql. Compute H = [hi i]; a qxq matrix
u·l = - R.B.P.x. = G.x .• l l J.. J. 1. 1
4)
5)
is dcterdned: ,je:{1, ••• q],
h~j= -e j
h
h~~
=
-J
h j =-
H
6)
0 else.
q+rq+j
= 111 11.:) L..
-Hr =
G-F.
Compute ~ ec $. ZIb-ac> } t" 1/t. -, l [li.. . A a "*L, '1'1\ . .J ,1\. . =1\. . - B . F. . i.j
; ..j
lJ _
lv
_
J-J
lJ
-1
l
lJ
eq
c=mrx\[Ii] ; ~ L.] (resp. AM l.] )
Ho
.... I
1\ q
= ~(A+BG), G =
b=mi n ~m [p i J ,:? i= Qj _+1' i Bi Ri Bi Pi
.
0
~
...
. - - ..... - .. - hr -. d'
h _ r 1 IT 0 'r The feedback matrix A~
( , , ••• ,u , )' ; u = u 1 + 1.1 2 , u 1=.u 1 k J = ,..,0 x'Px 0 ,P=diag (P 1 " " ' ? lC );61= IV =v(A+BG) ,G=dia gJG , ••• ,G ),A= 1 k A
fj ~2 ,
e q+ j
(B'B)-1 B 'A, u 2 =-Fx
A-A,
+e q+ + 0 rq j e q+r (q_1)+j e 1
8
Compvte F
v
Eo = [h~j]
Compute -1
f.= Al1 +q-l._J., iE $1, :t. "'n-J_ t ••• 1])
Solve the Riccati equation AiPi+PiAi-PiBiRiBiPi+Qi=O, ~i •
• • • , J~n 1 ' 0 , ••• ,0),
j€[q+1, ••• , ( q+1)r1; f=col (f 1 " •• ,fn+q)'
347
=lAc o
15) a) 'i;est a nd e-valuete step responb) se. If it is not satisfied choose A ' new, A -A new, go to 11) else go to end). 16) End. IJet us further forJl11Jlate Proble!n 2 solution. It is based on the Riccati algebraic subsystem solution and on the global control design minimizing the interconnection effect. 'i:he suboptimali ty index ~ evaluates the interconnection effect on the criterion value(r6J .1.The Riccati equation solution is a doubling continuous - time algori thnl effectively using a discre-
denotes the minimal (resp. maximal) eigenvalue of the corresponding matrix • 7)a) Test and evaluate step response, b) ~ , J ,
End.
The IBM 2250 display interaction is inclvded in two wa:'iTs. The step number of both al~orithms is followed by a) (resp.b»). a) (resp.b» means that the interaction in the problem data can occt1r (resp. the specified variables can be displayed as the time functions-step response).
The S8Hpling j.nterval din the both cases is ad justed on 0:-\1 (resp.J;\.2) in the Algorithm 1 (resp. Algorithm 2)
dA1 :. 0.5/ AM [A~] r1,
r1 .
0A2· 0.51 AM [A + BG 1
Of c01.1rseo~JA1(resp.J"~dA2) but by the grap..hical representation we can choose as:icT,i ts an integer. The IBM SSP/)60 and the user-simplified GS? for :PJJ 1 have been used.
348
L. Bakule
Example. Using the Algori tblTI 2 der.lonstrate the influence of t he state penalization lJsing Q on the s ubo ptimali ty indexJ. Let us con sider power systeM model formed by two cOl'.nected plants [2,6J. The considered system:
:~ 11 02.72 o
o
1 0 0 j'BI1= - (.!. 04 5 0 n -3.33 3.33 5. 2 C.I 12.5
,2 o 0J' 000 A" [~'72 000 0
R1= [1J •
000
t
Q1=
[Vi0 w0 00 goO]' 000 000
Vie;. 1. b.
A1=A2 ,B 1=B 2 ,A·12 =A 21 ,Q1=Q2 ,R 1=R 2 , xo=col(O 0
The corlputation ~. as been perforr::ed on the IBM 370/ 13 5 computer.
0.01 0 0 0 0 0).
For w=0.15 (resp.15; resp.150) the s uboptimality index ,=0.23 (resp. 0 .07;resp. 0 ). Fig.1a (resp.Fig.1b) denonstrates the freql1 ency deviatio!l for y/=0.15 (resp.w=15).By the increasing state p en alizatio ~!. the system is better daMped and logi c ally t he intercon~e ctio~ effect decreases. It practically vanisbes for ~&150.
t
Fig.1.a.
Af(Hz] (resp. A f ; resp • .A f 2 ; resp. t ls]) denotes thJ frequency deviation (resp. x 2 = .4 f 1 of the first subsystem, resp. x6 = A f2 of the second subsystem; resp. time). t max = 8ls1, d = 0.1 in all cases.
CONCJJUSION Two interactive procedures have beer described. The decentralized pole assj.gnme!lt problem "l nd the two-stage optimal cOlitrol desi.gn alGorithnlS. The simple use of the second procedure is given on the example. 1) Jl.nderson,B.D.O.(1 978). Second-order converge nt algorithms for the steady-state Ri ccati equation. Int.J.Control,vol.2 0 ,g,pp.2 95-306. 2) Bakule L. ( 1978 ) Two-level control Generation of nu ltiarea electric enerey systems. In I.Troch (Ed.), 0ir.1UletioYl of Control Sys tems, North-Holla nd ,pp. 161-1 GJ . 3) Bakule L.(1 979 ). On decentralized lerge-sca1e d:~lnBmj. c s~rstem control. In J.Bene~,L.Bakule (Eds.), 3rd Formator Symposium, Academia:--
205-218.
4) Brasch,P.H. and J.B. Pearson (1970). Pole plecenent using dyns@ic co~pen sators. IEEE AC ,vol.15,1,J 4- 4J . 5) Corfmat,J.!'. s i:d A. S. IV\orse (1975). Decentralized con trol of linear !!l1..11t1variable syster:s. Proc.6th I!:'i'.C ConGress, Boston, paper 43 .3. 6) Fosha,C. and O.L E13erd (1970). The r:!egawatt-freql1er.c~T control proble]!!. IEEE Pl\.;3,4,pp.%3-57 1 • 7) ~iljok,D.D., Bnd S.K. Sundaresha l1 (1976). OptirneIity vs. reliability. In Y.O.Ho, S. K. Mitter (Eds.), Directions in :r,arf?e 3c31e s ~stems, Plenum Press, Hew York,pp.1 4-152. 8 ) Wonham, W.M. (1974). 1inear multivariable control. A ~eo me tric approach. Springer Ver ag, Berlin.
Abstract.The paper deals with the interactive cOTIp ~ter aided control design alsori tlms for the continuous linear dynamic systems . The decentralized pole assiGr~1'1ent problerll for tne strongly connected suhsystems using the dynamic compensation a:::Jo. the t'No-stage optinal control desi gn al~orithD reliable under tbe structural pe rturbations of the interconnections are cOl".sidered.:rhe iilteractioll by the COTJputetio n concerns the system strl1cture Bnd tbe data specification, the eigenvalues specification and the Graphical representatio:: of the desired veriables as e step responses. The case study of the power system mode l is included. Ee;{\'lords. AIcori tl1nlS for digital computer control; interactive compllter aided desien, large-scale systems; mu l tivar:!_ab l e control; opti1:lal control. I N':"CP..ODTJ CT ION
There i s increasing i n terest in tbe large-scale sys tem Llethodolog::r . A large-scale system may be considered as an interconnection of the s u bs~rste!:1s. ~he co~plexitv of the svstem is nain ly dete~mined - by t ~ ~ in~erconnections . A desirable feature is the separation at the subsyste~s a n ~ the i~te~con nection level. ':'he :::ost 13:i.nple approach is to ne~lect the interconnections and treat t~e lower order s ubsystems only. J uch a n approach works well if there is little interaction occ uring between controllers. Let us deal VIi th two procedures for the control design of t he large-sc/)le linear d;,rnaT!1ic s ysteDs, \'lhe re th.e i ::1terac tior.s infl1..1el:ce Ol·, t h0 s:',Tste r,~ activity. ~hc dcca n tr3li3ed polo ass:i_cr.!lellt proced :.~ ro cleels ;-:i th t~l.C c1:'nan~,c co:-.rpel~satio :: desiGn for the controllers so that t ~ e closed-loop eic;envalues 2~e prescribe d for the k -cha::1nel , j ointl~ co n trollable, jointly observa ble, si;ron~ly connected s~rsteY.ls [ 5]. :;"11e co nce ~t of t~~e cO:',lplcte s:rster.: is 1.' sed to :Ja l:e the systen bot~ co ntrollable and observa b le throl'3> a s inJle c::a~c -,-, e I b:r t:'-~e applicatioE of t~le :".oi:d:,r:1a:-:ic dece :1tralized feedbac :: to 011 c:Oan':e1s . Then t he Bra sc ~ - ?earso ~ procedure [4Q ca ~
be
't~sed
to construct t :J e
The two -stage optimal control procedure is s upplied by the power system Dodel control del3iGn example.
JJet us fori:iulate the froblern and the l-'ro b J.elii :2 . Problem 1. The system 3 is described as l:iJleer k-ch arcnel system of a form
',Jr=Ax+t 3.u., i= 1 1. 1. ~Ti = Cix, ., WDe r~.
xl:
';'!D \1.
i=1, ••• ,k~2, .
(resp.
11
1li.€ i
(1)
rH.
~
; resp.
y.£R ~ ) is t~e sta te (resp . the co~ ~
trol, resp. t )1 0 ont p,; t) vector. AE. f{n x n , I\€ Rn x nli , C € R !J i x n i are the consta~t matrice s describinG the systefl d;pa mi!:: s. Le t us suppose that S is a joint ly co~tro lla ble , j ointly obs ervable,i .e.
d:n.1 a l~-:i c
lo ca l control resulti ~G in a Cl0BGdJ_oop systen \',i t1'2 prescribed spectr1l!n.
S.F.(,.('. - M*
The two-staGe optimal co n trol desi~n proc ed1.1 re reliable under the str'.1ctl1 ral perturbati ons.Such perturbation C9n resul"; in i :::.stabj.li ty of the S;,TStem ['7J. The fi_rst stage deals \':1 to the local control desien, the second stage deals with the cont rol design minimizing the interconnec tion effect. Its i n fluence o~ the cost criterion is eva l119ted by t?J.e CN1PlltillG of the Sllboptil'!1ali ty index (7).
345
346
L. Bakule
the triple (e,A,B), C = col (C 1 ' ••• ••• ,c1,),:8 ~ ~O\" (B1"",l):) is cor.Jplete15 ,8J.(.) means transpose (.). decentralized pole assignment p!'oblen is to find the dynam:Lc local controllers of c form
~'he
.
zi
=
Sizi + RiYi'
Uin~ Qizi + FiYi' Z~E.TI.
i
= 1, ••• ,k (2)
so thet the closed loop spect.rum of the system (1),(2) is freely assiGnable. Note that the nondynamic loc::ll control has a for]"! l
1.\
= FiYi
min ,T
(4)
U
s\.lbject to
,
' E T) n.x n. -.., ".,·,m. x r.J. n >0 ,./; le l l,hi,,"n l l , ·oi- ,
~
n.
ts both cOY'.trollable and observable t~lrOl.lgh a si::Jgle channel denoted .;i. '1'he11 the dynanic conpensator can De applied to construct a control closedloop systen with prescribed spectrum [~. Theorem 1 (resp. Corollary 1) SOlve this problem [SJ.I'he triple (C,A,B) is complete if (A,B),(A,C) is controllable, observable[s]. n
,
Dj
.
Ri>O; XiE:R l i, resp. uiE:H -) lS the state vector (resp. the control).'l'he system dynamics is described by the constant Elstrices Al' ,B-;... ,A; -;. I,et us suppose k peirs (Ai,n i ) cont!'olable.
n+,.
4C
uj.n
11 A
BFII ,
}3
1
Let us suppose the control D = 1} + 1.11 •
(7)
;3CJIilJIn OH - ALGOEITIIIVI3
I ·et ,'S fo:.~r.;uJ.[lte t~1 e alsori tl1ms on the basis of the knoqn res ults. The solution consists of the solution of Problem 1 (resp. Problem 2) as the Algorithm 1 (resp. Algorit hm 2). The basic approacb to the Problen 1 solution is to apply first local nondynamic controls (J) so tllSt the closed loop system
'"
IIAII~II>B·tl·r.;·II;A '[' l l l c
4)
=
=A +lB.H.r: - - l l.• If it is satisfied Tgo~to 4) else go to end). If j = k-1 go to 5) else go to 3) •
i£(1, ••• ,k} A = A ,n = B., C = C .• c 'it' a-
5) a) Choose
6)
- Q U~ G U~ 1 ' • • • , Al ), = c1i8ij ( B , ... , I\J. (
tJn",/l.. =1
where ~o (resp. ~c) denotes the observability index (resp. the controllability index) [4,5]. Algorithm 1. 1) a) Give A,Bi,Ci,~i,j=O,xo = x(t o ) 2) Test ths joint controllability, joint observability (C,A,]), .£ = (C1"",Ck)'~ =~TIi • If it is not satisfied go to end) • J) j = j + 1. Test the completness (Cj,Ac,B j ) using the pseudorandom generator. So that
(6)
·..,here J,. l\..
.
K.!: ,a 0 ).'f3i
the desired set, cenerally complex conjugate pairs, of the eigenvalues A = [ .A~, ... ,).n+1-1; t- min f ~Ol ale},
~<.l
I,et us find the control optimizing ooth the stlbsystems Beti vi ty and the i nfltiGl1ce of tbe L1terconnections. ']'hei:: find F
.
Denote: I.AI - AI =.[ ).1 ~i the · desired ,-0 characteristic polynomial
(3)
resulting from (2) for n i = O. :Get us formllate the Problem 2 now. Problem 2. Let us ~ind the control fi for the cost
VI.~lGre
(8)
Test the A-cyclic property. If it is not cyclic go to 7) else go to 8). Using the pseugo-random genera.7) tor geDerete H 52 that U,,'..II=IIBHclI, A:=A+BHC; c.;o to 6). G) a) Choose c' = d' C so that (A,c') is ohservable. 9) Compute a column vector :L. I = [L i]
i= 1, ••• , n;
- d'A i - 1 n iri J d'A n - k - 1B·, 1";~+'-~ k -
10 )
ka f - -
Compute q. 11)8) Give A. 12) Solve equation Pe = f, r= m~, ". w~lere e (resp.f) is q+r(q+1)
Large scale system control design
(resp. n+q) column vector, F=[Pij]; Pj=col (P1j"" ,Pn+q, j ); Pj=col( O, ••• ,O,p .'j =1, ,(11-1' ••• , «()' 0, ••• , 0) , j £ [1 , ••• , Pj=col (0 , ••• , 0 , Pjj=L 11 ,L21""
qJ ;
te-time problem reletion [1J. I.et us ~ l rther suppose IBI = m. Algorj. thm 2. 1)8) Give ,\,Bi,Aii,Qi,Ri,i"i,xo=x(to)' tJ
2)
3)
,nJ
fi= ;1q_i_n,iE [n+1, ••• ,n+ql. Compute H = [hi i]; a qxq matrix
u·l = - R.B.P.x. = G.x .• l l J.. J. 1. 1
4)
5)
is dcterdned: ,je:{1, ••• q],
h~j= -e j
h
h~~
=
-J
h j =-
H
6)
0 else.
q+rq+j
= 111 11.:) L..
-Hr =
G-F.
Compute ~ ec $. ZIb-ac> } t" 1/t. -, l [li.. . A a "*L, '1'1\ . .J ,1\. . =1\. . - B . F. . i.j
; ..j
lJ _
lv
_
J-J
lJ
-1
l
lJ
eq
c=mrx\[Ii] ; ~ L.] (resp. AM l.] )
Ho
.... I
1\ q
= ~(A+BG), G =
b=mi n ~m [p i J ,:? i= Qj _+1' i Bi Ri Bi Pi
.
0
~
...
. - - ..... - .. - hr -. d'
h _ r 1 IT 0 'r The feedback matrix A~
( , , ••• ,u , )' ; u = u 1 + 1.1 2 , u 1=.u 1 k J = ,..,0 x'Px 0 ,P=diag (P 1 " " ' ? lC );61= IV =v(A+BG) ,G=dia gJG , ••• ,G ),A= 1 k A
fj ~2 ,
e q+ j
(B'B)-1 B 'A, u 2 =-Fx
A-A,
+e q+ + 0 rq j e q+r (q_1)+j e 1
8
Compvte F
v
Eo = [h~j]
Compute -1
f.= Al1 +q-l._J., iE $1, :t. "'n-J_ t ••• 1])
Solve the Riccati equation AiPi+PiAi-PiBiRiBiPi+Qi=O, ~i •
• • • , J~n 1 ' 0 , ••• ,0),
j€[q+1, ••• , ( q+1)r1; f=col (f 1 " •• ,fn+q)'
347
=lAc o
15) a) 'i;est a nd e-valuete step responb) se. If it is not satisfied choose A ' new, A -A new, go to 11) else go to end). 16) End. IJet us further forJl11Jlate Proble!n 2 solution. It is based on the Riccati algebraic subsystem solution and on the global control design minimizing the interconnection effect. 'i:he suboptimali ty index ~ evaluates the interconnection effect on the criterion value(r6J .1.The Riccati equation solution is a doubling continuous - time algori thnl effectively using a discre-
denotes the minimal (resp. maximal) eigenvalue of the corresponding matrix • 7)a) Test and evaluate step response, b) ~ , J ,
End.
The IBM 2250 display interaction is inclvded in two wa:'iTs. The step number of both al~orithms is followed by a) (resp.b»). a) (resp.b» means that the interaction in the problem data can occt1r (resp. the specified variables can be displayed as the time functions-step response).
The S8Hpling j.nterval din the both cases is ad justed on 0:-\1 (resp.J;\.2) in the Algorithm 1 (resp. Algorithm 2)
dA1 :. 0.5/ AM [A~] r1,
r1 .
0A2· 0.51 AM [A + BG 1
Of c01.1rseo~JA1(resp.J"~dA2) but by the grap..hical representation we can choose as:icT,i ts an integer. The IBM SSP/)60 and the user-simplified GS? for :PJJ 1 have been used.
348
L. Bakule
Example. Using the Algori tblTI 2 der.lonstrate the influence of t he state penalization lJsing Q on the s ubo ptimali ty indexJ. Let us con sider power systeM model formed by two cOl'.nected plants [2,6J. The considered system:
:~ 11 02.72 o
o
1 0 0 j'BI1= - (.!. 04 5 0 n -3.33 3.33 5. 2 C.I 12.5
,2 o 0J' 000 A" [~'72 000 0
R1= [1J •
000
t
Q1=
[Vi0 w0 00 goO]' 000 000
Vie;. 1. b.
A1=A2 ,B 1=B 2 ,A·12 =A 21 ,Q1=Q2 ,R 1=R 2 , xo=col(O 0
The corlputation ~. as been perforr::ed on the IBM 370/ 13 5 computer.
0.01 0 0 0 0 0).
For w=0.15 (resp.15; resp.150) the s uboptimality index ,=0.23 (resp. 0 .07;resp. 0 ). Fig.1a (resp.Fig.1b) denonstrates the freql1 ency deviatio!l for y/=0.15 (resp.w=15).By the increasing state p en alizatio ~!. the system is better daMped and logi c ally t he intercon~e ctio~ effect decreases. It practically vanisbes for ~&150.
t
Fig.1.a.
Af(Hz] (resp. A f ; resp • .A f 2 ; resp. t ls]) denotes thJ frequency deviation (resp. x 2 = .4 f 1 of the first subsystem, resp. x6 = A f2 of the second subsystem; resp. time). t max = 8ls1, d = 0.1 in all cases.
CONCJJUSION Two interactive procedures have beer described. The decentralized pole assj.gnme!lt problem "l nd the two-stage optimal cOlitrol desi.gn alGorithnlS. The simple use of the second procedure is given on the example. 1) Jl.nderson,B.D.O.(1 978). Second-order converge nt algorithms for the steady-state Ri ccati equation. Int.J.Control,vol.2 0 ,g,pp.2 95-306. 2) Bakule L. ( 1978 ) Two-level control Generation of nu ltiarea electric enerey systems. In I.Troch (Ed.), 0ir.1UletioYl of Control Sys tems, North-Holla nd ,pp. 161-1 GJ . 3) Bakule L.(1 979 ). On decentralized lerge-sca1e d:~lnBmj. c s~rstem control. In J.Bene~,L.Bakule (Eds.), 3rd Formator Symposium, Academia:--
205-218.
4) Brasch,P.H. and J.B. Pearson (1970). Pole plecenent using dyns@ic co~pen sators. IEEE AC ,vol.15,1,J 4- 4J . 5) Corfmat,J.!'. s i:d A. S. IV\orse (1975). Decentralized con trol of linear !!l1..11t1variable syster:s. Proc.6th I!:'i'.C ConGress, Boston, paper 43 .3. 6) Fosha,C. and O.L E13erd (1970). The r:!egawatt-freql1er.c~T control proble]!!. IEEE Pl\.;3,4,pp.%3-57 1 • 7) ~iljok,D.D., Bnd S.K. Sundaresha l1 (1976). OptirneIity vs. reliability. In Y.O.Ho, S. K. Mitter (Eds.), Directions in :r,arf?e 3c31e s ~stems, Plenum Press, Hew York,pp.1 4-152. 8 ) Wonham, W.M. (1974). 1inear multivariable control. A ~eo me tric approach. Springer Ver ag, Berlin.