Near-Optimal Feedback Control
Distributed Systems with Distributed Controls by Trajectory Approximation A near-optimal feedback law for a distributed system with a distributed control and a quadratic performance index i s determined. The method used, trajectory approximation, i s based on the reduction to a system of lumped eqvations b y the method of weighted residuals. An application of Pontryagin’s maximum principle leads to a matrix Riccati equation whose solution yields the desired near-optimal feedback law. Numerical results obtained for a plug flow heat exchanger with a spatially distributed wall flux as the control indicate that optimality i s attainable with three or four sampling probes.
R e c e n t atteiitioii to di,qtributed paraiiieter systeiiis has rebultetl iii the forniulatioii of the iiecesbary and sufficieiit c*oiiditioiis for o1)tirnal coiitrol for a n-icle claw of 1irolilcins (Hutkovskii, 19Gla,b; Ileiin, 1968; Egorov, 1964; Iio1qiel et ab., 1968). In certaiii iiistaiices it i b ~)os>ilileto reprrwit the o1)tiiiial coiitimol Iiolicies in teriiis of feedback l a w (Ko1q)eI :tiid Shih) 19Gti). Such larvb geiierally have Iieeii discovered oiily i‘ol, liiicnr 1xirti:il differeiitial equatioiis aiid quadratic pci~foriii:iiicc criteria, h i t repre,-eiit :iii effective iiieaii.. of iiiil)lciiiciitiiig olitimnl control. Such f o ~ i i i u l a t i ~ i i isuffer s froiii two dra\vl)ncks wliicali i n n > 1)rovc>iit tliclir i,e:tdy iiiil)leriieiit:itioii. The firht i h that t h e tqu:itioiis froiii ivliich the feedl)ack gaiii i q to lie coiiil)utetl arc c~iiiililic,:itctl1)aitial differciitial-iiitegral equatioiiq, n.lio%c ioliitioii iiiiibt 1)c olitniiied i i i :in iteratiw \va!.. ‘The secoiid d r : i ~ l ) a ( ,i*k that the fecdl)nck law i. esl)i,essetliii terms of thc b t : i t ( , varialilc at evei’y ,oiii(>f ~ i of~ :iii])i’osiiii:itioii i of the o1)tiin:il feetlhc~kcoiiti,ol law iiiii-t h e iiiadr, For esaiiil)le, rclilaccriieiit of the 1)nrtial diffci~clitinlequ:itioiib by fiiiite tliffci,ciic~:q)l)rosiiiiatioiis has Iiecii .uggt>tctl :xi : Iiiicthotl for Iimidliig distrilnited 1iar:inicter o1)tiiiiiziitioii prol)leiiisJ aiitl hoth tlirory aiid e m i i i 1 ) l r b cwi 1)c forillti iii l h r liternturc (Ihltkov-kii, 1963; Gr:ih:im :tiid I)’douz:1, 1969). T h e c,oiiiliut:itioit of iieai~-o~itiiid control 1)oliciex I J ])yo~ ceclures other thaii finite differencing has received far less atteiitioii, iii bliite of the fact that, iii niaii>- iiistances, other tecliiiiqucb for \olviiig IimtiiiI differeiitial equations are equally effective. I11 pai,ticulai,,byiithesis techiiique,*,based 011 geiiernlizetl iiietliotl; of weightctl residuals, have fouiid esteiisive II‘C iii citgiiieeriiig al)l)licatioiis (Kal)Iaii, 1967 ; Yasiiisky, 19GG) aiitl iiiiclear reactor desigii (Ii:~l)lail,1967). Receiitl?. I’arkiii :iiid Zahratlnik (1970) and Zahradiiik aiid Lyiiii (1970) deinon 0 are n-eightiiig factors which rnap iii geiicral depend both 011 time and position. To deteriiiiiie the near-olitiiiial feedback policy, the state varialile i q alq)roxiiiiated iii the folloxiiig way,
s x ( r , t ) = .?(r,t)
a j ( t ) P j ( r )= aTP
=
(3)
j-1
where a represeiitq the K-dimensional vector of time-depeiideiit mixiiig coefficients aT =
[aI,a?,.. . , a . v l
aiid P is a vector made ul) of X members of a complete set of fuiictioiis for rt[O,l] which satisfy Pt(0) = O ( i = l,,. . ,.V). The control effort is similarly approximated as 711.
u(r,t) = z i ( ~ ! t = )
Cl(t)Sj(r= ) Cr(t)S(r) j=1
(4)
whcrc 7 - T = [c'1772 . , T I n i ] represents an m-dimensional vector of tiiiie-dei)tndeiit controls and S ( T ) consi.;ts of nz members of :I cornl)lete set of functioiii on [0,1]. The S j need not be ideiitical to t h e Pj. The :il)l)rosimntioiis for x ( r , t ) aiid u(r,t) are substituted iii 1.Cqii:itioii 1 and the re5iduali are required to be orthogonal to :I >ct of weighting functioiis w ( r ) p F , ( rin ) the following fa-hioii (W'&,tuTP
+ aTP'
- 17TS) = 0
(j
=
1,2. . , , L V )
The syinliol ( f 3 g ) re1)meiits the inner product
EXACT A PPROXI MAT E
0.8
(5)
L1
j(r)g(r)dr
for c o n t i i ~ ~ i ofiinctioiih ~~s f aiid g on [0,1], w ( r ) is a strictly Imitive weighting function, aiid the p i ( r ) may be either the P , ( r ) or a \et of liol~minialsnearly orthogoiial to the Pi(r) on [0,11. The dot :ind prime denote time aiid space derivatives, re+l)ectively, Equ:~tioii 5 coiistituteb a .et of .Y ordinary differential cquatioiis in cy wliirli niay lie written
tu
=
+ Br-
.-la
(6)
=I = II-lI, :iiitl B = II-l.lI. If and -11 are 5 x .Y iii:itri~c>:iiid III i q :in S x TU matrix, n-hose cornpoileiits arc
0.2
0 DISTANCE, L
Figure 1 . Optimal and near-optimal distributed controls fort, = 0.5
wlicw
I!ij = (Wpz,Pj)
li,
= (WC.j,
VI13 =
--PJ')
(w(Ct,S,)
S u l i ~ t i t i i t i ~ofi i ,?(r,t) aiid a(r,t) into Eqiizitioii 2 yield$ the aiqirosiniatioii
If the s1i:itinl iiitcgratioii i q carried out, J ( u ) may be written as J ( z c )=
i Lt(
[a'Q(t)a
~-
+ r-TR(t)C.]dt
(7)
whwe Q ( t ) is a n S x S 1)ositivc seiiiidefinite syninietrir matrix : i d R ( f ) is a n vi x vi positive definite syniiiietric matrix, with coni1)onent~
udt)
=
bPf,PJ?
ri,(t)
=
(pSi,Sj)
=
-R-'(t)BTK(t)a(t)
(8)
where K ( t ) is the solution of the matrix Riccati equation
R(t) =
-K(t)d
- -4TK(t) + K ( t ) B R - ' ( t ) B ' K ( t ) - Q ( t )
satisfying K(t,) = 0. Equation 8 is not quite in the forni of a feedback law, but it can be made so in t h e following n a y . Suppose t h a t A' sensors or robes are located a t discrete spatial locations, r I I r 2. . . ,r.\-. The state variables at these locations may be approsimated as
x ( r ( > t )= cyT(t)P(ri) If
xJb(t) = GCY(t) where G is a n S
x &Ymatrix whose coefficients are QlJ
=
P,Jrt)
where k T ( r J t ) = S*(r)R-l(t)BTK(t)G-l is the near-optinial feedback gain vector. The abilitl- of the near-optinial feedback coiitrol law (Equation 11) to match t h e optiiiial one is illustrated by t h e application of trajectory approsimatioii to a 1)lug flow heat eschanger with t h e wall flux ai: the distributed control. Computational Results for a Tubular Heat Exchanger
T o illustrate the method, c o i i d e r a perforniaiice index in which p is a constant and p = 1, =
1 L';sb [px2(r,t)+ u*(r,t)]drdt 2
(9)
(12)
Koppel and Shih (1968) have treated this prolileni in some detail for the initial condition x(r,O) = r. The simple iiature of the problem permits the solution of Equations 1 and 12 in closed form. The application of a strong miiiinium principle due to Kolipel and Shih rewlts in t h e following equatioiis for t h e optimal control, ic, aiid profile f . tl(r,t) =
and
-A.x(r,t) taiih [A(1 - r ) ] , t < T, (1321) -As(r,t) tanh [ A ( t , - t ) ] , 7 5 t < r , t , < 1, (13b) J t> r (13c)
i o i o
- t ) cosh [h(l - r ) ] bech [ A ( l - r ( r - t ) cosh [ A ( t , - l ) ] sech (At,) (T
f(r,t)=
,-
deiiotes the Lv-\-ectorof st'ate variables a t the locations . ,r.ye[O,l], it is related to a ( t ) iii the followiiig way
S/b(t)
rl,T2,. ,
.(t) = G-'.ZIb(t) (10) Substitution of Equations 8 and 10 into Equation 4 yields the near-optimal feedback law
J(u)
Tlie 1irol)leni liosed by Equations 6 aiid i is to obtain the control liolicy \%-hirhniiiiiniizes J ( u ) .Since thc :ind Iwioriiiunce cariterion quadratic, ail ol)tiinal feedback law of the followiiig form esiqts (Althaiisand Falb, 1966) 7.(f)
Since the r i are discrete, G has an inverse, so that it is possible t o solve Equation 9 for
+ t)]
(l4a) (14b) (14c)
-+
where A = d p , 7 = f , r - 1, and Equations 14a, 14b, and 14c hold for the same conditions 011 t and T as Equatioiis 13a, 13b, aiid 13c, resliectively. The results for zi(r,t) with t , = 0.5 and 1 and p = 5 are plotted as a dashed curve iii Figures 1 and 2. To denionstrate t h a t trajectory approximation leads to physically useful results aiid t o provide a coinliarison with Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
177
Koppel aiid Shih's (1968) inetlioti, coinputatioiis were performed oil the L-?JIT~.4C-1108using the saine value> of p = 5 and t , = 0.5 and 1. The trial fmictioiis were taken to be S j ( r ) = r j ( j = 1,2,. . .,.V). S , ( r ) = r?(,j = 1 , ., ,,VI),and the Galerkiii ~veightingfiinctioiis used were w ( r ) = 1 and q i ( r ) = Ti-l*(r) (i = 1,.. .!.Y). The are the Chebyshev polyiioiiiial.~of the firht kind, which are orthogonal with respect t o ( T - T * ) - ~ , ' on ~ [0,1]. .'\fter coniliuting matrices A , B , Q, and R , the niatris Riccati equation with K(t,) = 0 was integrated liackn-ard in time u.:ing a fourth-order RuiigeIiutta subroutine. Comliutatioii time was about 4 seconds f o r S = 1. The iiear-olitiinal feedback gain vector k(r,t) in Equation 11 was computed for t , = 1 and it.; eo~npoiie~its are plotted in Figure 3 as a function of t for iV = 3 for the three probe and ra = b, 6. 111 Figure 4, the locatioiii: rI = , 6, r2 = coinlionent.; of k are plotted agaiwt r at time t = 0. The 1irol)c locatioiii were chobseiiai: reasonable values; if a different set of locatioiia i-: u v d , the new feedhack gain vector k l ~ ( t ) may be computed from
The iiive4gatioii of trajectory aliproxini:itioii foi, thc tiilnilur heat exchanger ivaq conilileted by calculating a1q)roxiiii;itioiis to the near-optininl control n i t ) froin Equatioli 11, using K ( t ) anti values of .rf(,(t) coiiil~tctifroin Equatioiiq 14. The near-o~itinialcontrol.. thu-: ol~taiiietiare ~ilottetla. -olid curves iii Figure- 1 aiid 2 for ,U = 5 mid tf = 0.5 aiid 1. Discussion
The technique of trajectory a~qiroxiiiintioiiha.; I)eeii bhon.11 to be ail effective ineaiis of coin1)utiiig 1iear-01)tiinal fecclliack tein.; deqc~ribedby first-oi,der liiicar liy1)crbolic equations. 13). the retluctioii of distributetl ~iaraincter equation> to :I systein of luinlied 1)nrarneter equatioiis, the necessity of holviiig a partial differelitid equiitioii i\ hypas,-etl, ant1 moreover, the l a q e l~otlyof theory for the o~)tiin:ilcontrol of lumped l~arainctcrequatioii-: i\ Inade ovai1al)le. -4iiother feature of trajector!. approsini:itioii, a' iii all apiilications of the ~iietliotlof n.eightcd re>idu:il>>is the ability to bring into pl:t>. 1)hysic.d iii-ight vix tlic choicr of t i i d fuiictioli>. The trial fuiictioiis, ri, were clio-ell for coiiv~iiieiice only, The use of other trial function.: could perhaps lead to the same degree of wccesi n.ith maller value.; of .V than those used i i i the calculatioii+ fol the heat excthaiigcr.
klqt) = LT(~)GG~-~ where GI in the geonietry inatris for the new probe locations.
1.0
_-
1
-EXACT APPROXIMATE tf =I0 \
I
/
08
3 I
06
-I
0 [r
!j0.4 0 V
0.2
0 02
04
08
06
I O
DISTANCE, h
Figure 2.
Optimal and near-optimal distributed controls for t, = 1
I4l---A_-
+
. --. ,
12
,
-,"a= Y6
--
- - " lh
\-
tf.10 N=3
y6
---,?,=
I O
)X
0 8
z -
a w
06
Y
V
